[ { "id": "Differential_equations_0000", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "3", "keywords": [ "differential equation' 'solution" ], "problem_v1": "Find all values of $k$ for which the function $y=\\sin(kt)$ satisfies the differential equation $y''+16 y=0$. Separate your answers by commas. [ANS]", "answer_v1": [ "(4, -4, 0)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find all values of $k$ for which the function $y=\\sin(kt)$ satisfies the differential equation $y''+3 y=0$. Separate your answers by commas. [ANS]", "answer_v2": [ "(1.73205080756888, -1.73205080756888, 0)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find all values of $k$ for which the function $y=\\sin(kt)$ satisfies the differential equation $y''+7 y=0$. Separate your answers by commas. [ANS]", "answer_v3": [ "(2.64575131106459, -2.64575131106459, 0)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0001", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "3", "keywords": [ "differential equation' 'initial" ], "problem_v1": "The family of functions y=ce^{-2x}+e^{-x} is solution of the equation y'+2y=e^{-x}. Find the constant $c$ which defines the solution which also satisfies the initial condition $y(3)=6$. $c=$ [ANS]", "answer_v1": [ "2400.48722403322" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The family of functions y=ce^{-2x}+e^{-x} is solution of the equation y'+2y=e^{-x}. Find the constant $c$ which defines the solution which also satisfies the initial condition $y(-5)=10$. $c=$ [ANS]", "answer_v2": [ "-0.00628394770146062" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The family of functions y=ce^{-2x}+e^{-x} is solution of the equation y'+2y=e^{-x}. Find the constant $c$ which defines the solution which also satisfies the initial condition $y(-2)=7$. $c=$ [ANS]", "answer_v3": [ "-0.00712581101547343" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0002", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "3", "keywords": [ "differential equation' 'solution" ], "problem_v1": "Match each differential equation to a function which is a solution. FUNCTIONS A. $y=3x+x^2$, B. $y=e^{-6x}$, C. $y=\\sin(x)$, D. $y=x^{\\, \\frac{1}{2} }$, E. $y=7 \\exp(6x)$, DIFFERENTIAL EQUATIONS [ANS] 1. $y''+12 y'+36 y=0$ [ANS] 2. $y''+y=0$ [ANS] 3. $2x^2y''+3xy'=y$ [ANS] 4. $y'=6 y$", "answer_v1": [ "B", "C", "D", "E" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Match each differential equation to a function which is a solution. FUNCTIONS A. $y=3x+x^2$, B. $y=e^{-8x}$, C. $y=\\sin(x)$, D. $y=x^{\\, \\frac{1}{2} }$, E. $y=4 \\exp(3x)$, DIFFERENTIAL EQUATIONS [ANS] 1. $y'=3 y$ [ANS] 2. $y''+10 y'+16 y=0$ [ANS] 3. $xy'-y=x^2$ [ANS] 4. $y''+y=0$", "answer_v2": [ "E", "B", "A", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Match each differential equation to a function which is a solution. FUNCTIONS A. $y=3x+x^2$, B. $y=e^{-6x}$, C. $y=\\sin(x)$, D. $y=x^{\\, \\frac{1}{2} }$, E. $y=5 \\exp(3x)$, DIFFERENTIAL EQUATIONS [ANS] 1. $y''+9 y'+18 y=0$ [ANS] 2. $y''+y=0$ [ANS] 3. $y'=3 y$ [ANS] 4. $2x^2y''+3xy'=y$", "answer_v3": [ "B", "C", "E", "D" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Differential_equations_0003", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "3", "keywords": [ "differential equation' 'initial", "Initial value problem" ], "problem_v1": "It is easy to check that for any value of c, the function y=x^2+ \\frac{c}{x^2} is solution of equation xy'+2y=4x^2,\\ (x > 0). Find the value of $c$ for which the solution satisfies the initial condition $y(8)=6$. $c=$ [ANS]", "answer_v1": [ "-3712" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "It is easy to check that for any value of c, the function y=x^2+ \\frac{c}{x^2} is solution of equation xy'+2y=4x^2,\\ (x > 0). Find the value of $c$ for which the solution satisfies the initial condition $y(1)=10$. $c=$ [ANS]", "answer_v2": [ "9" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "It is easy to check that for any value of c, the function y=x^2+ \\frac{c}{x^2} is solution of equation xy'+2y=4x^2,\\ (x > 0). Find the value of $c$ for which the solution satisfies the initial condition $y(4)=7$. $c=$ [ANS]", "answer_v3": [ "-144" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0004", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "2", "keywords": [ "differential equation' 'system" ], "problem_v1": "Just as there are simultaneous algebraic equations (where a pair of numbers have to satisfy a pair of equations) there are systems of differential equations, (where a pair of functions have to satisfy a pair of differential equations). Indicate which pairs of functions satisfy this system. It will take some time to make all of the calculations.\ny_1'=y_1-2 y_2 \\qquad y_2'=3y_1-4 y_2 [ANS] A. $y_1=\\sin(x) \\qquad y_2=\\cos(x)$ B. $y_1=e^x \\qquad y_2=e^x$ C. $y_1=\\cos(x) \\qquad y_2=-\\sin(x)$ D. $y_1=2e^{-2x} \\qquad y_2=3e^{-2x}$ E. $y_1=\\sin(x)+\\cos(x) \\qquad y_2=\\cos(x)-\\sin(x)$ F. $y_1=e^{4x} \\qquad y_2=e^{4x}$ G. $y_1=e^{-x} \\qquad y_2=e^{-x}$\nAs you can see, finding all of the solutions, particularly of a system of equations, can be complicated and time consuming. It helps greatly if we study the structure of the family of solutions to the equations. Then if we find a few solutions we will be able to predict the rest of the solutions using the structure of the family of solutions.", "answer_v1": [ "DG" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "Just as there are simultaneous algebraic equations (where a pair of numbers have to satisfy a pair of equations) there are systems of differential equations, (where a pair of functions have to satisfy a pair of differential equations). Indicate which pairs of functions satisfy this system. It will take some time to make all of the calculations.\ny_1'=y_2 \\qquad y_2'=-y_1 [ANS] A. $y_1=e^x \\qquad y_2=e^x$ B. $y_1=e^{4x} \\qquad y_2=e^{4x}$ C. $y_1=\\sin(x) \\qquad y_2=\\cos(x)$ D. $y_1=\\sin(x)+\\cos(x) \\qquad y_2=\\cos(x)-\\sin(x)$ E. $y_1=2e^{-2x} \\qquad y_2=3e^{-2x}$ F. $y_1=\\cos(x) \\qquad y_2=-\\sin(x)$ G. $y_1=e^{-x} \\qquad y_2=e^{-x}$\nAs you can see, finding all of the solutions, particularly of a system of equations, can be complicated and time consuming. It helps greatly if we study the structure of the family of solutions to the equations. Then if we find a few solutions we will be able to predict the rest of the solutions using the structure of the family of solutions.", "answer_v2": [ "CDF" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "Just as there are simultaneous algebraic equations (where a pair of numbers have to satisfy a pair of equations) there are systems of differential equations, (where a pair of functions have to satisfy a pair of differential equations). Indicate which pairs of functions satisfy this system. It will take some time to make all of the calculations.\ny_1'=y_2 \\qquad y_2'=-y_1 [ANS] A. $y_1=\\sin(x) \\qquad y_2=\\cos(x)$ B. $y_1=e^{-x} \\qquad y_2=e^{-x}$ C. $y_1=2e^{-2x} \\qquad y_2=3e^{-2x}$ D. $y_1=\\cos(x) \\qquad y_2=-\\sin(x)$ E. $y_1=e^x \\qquad y_2=e^x$ F. $y_1=e^{4x} \\qquad y_2=e^{4x}$ G. $y_1=\\sin(x)+\\cos(x) \\qquad y_2=\\cos(x)-\\sin(x)$\nAs you can see, finding all of the solutions, particularly of a system of equations, can be complicated and time consuming. It helps greatly if we study the structure of the family of solutions to the equations. Then if we find a few solutions we will be able to predict the rest of the solutions using the structure of the family of solutions.", "answer_v3": [ "ADG" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Differential_equations_0005", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "3", "keywords": [ "initial value' 'differential equation", "Initial value problem" ], "problem_v1": "The solution of a certain differential equation is of the form y(t)=a \\exp(5 t)+b \\exp(9 t), where $a$ and $b$ are constants. The solution has initial conditions $y(0)=4$ and $y'(0)=4.$ Find the solution by using the initial conditions to get linear equations for $a$ and $b.$\n$y(t)=$ [ANS]", "answer_v1": [ "8 *(e^(5 *t)) + -4 *(e^(9 *t))" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The solution of a certain differential equation is of the form y(t)=a \\exp(7 t)+b \\exp(8 t), where $a$ and $b$ are constants. The solution has initial conditions $y(0)=1$ and $y'(0)=2.$ Find the solution by using the initial conditions to get linear equations for $a$ and $b.$\n$y(t)=$ [ANS]", "answer_v2": [ "6 *(e^(7 *t)) + -5 *(e^(8 *t))" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The solution of a certain differential equation is of the form y(t)=a \\exp(5 t)+b \\exp(7 t), where $a$ and $b$ are constants. The solution has initial conditions $y(0)=2$ and $y'(0)=3.$ Find the solution by using the initial conditions to get linear equations for $a$ and $b.$\n$y(t)=$ [ANS]", "answer_v3": [ "5.5 *(e^(5 *t)) + -3.5 *(e^(7 *t))" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0006", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "2", "keywords": [ "differential equation' 'higher order" ], "problem_v1": "If $L=D^2+5x D-4x$ and $y(x)=4x-4 e^{3x},$ then $Ly=$ [ANS]", "answer_v1": [ "20 x - 16 x^2 - 36 e^{3 x} - 44 x e^{3 x}" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "If $L=D^2+2x D-5x$ and $y(x)=2x-3 e^{5x},$ then $Ly=$ [ANS]", "answer_v2": [ "4 x - 10 x^2 - 75 e^{5 x} - 15 x e^{5 x}" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "If $L=D^2+3x D-4x$ and $y(x)=3x-4 e^{2x},$ then $Ly=$ [ANS]", "answer_v3": [ "9 x - 12 x^2 - 16 e^{2 x} - 8 x e^{2 x}" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0007", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "2", "keywords": [ "ivp' 'initial' 'value' 'problem" ], "problem_v1": "The differential equation\ny^{\\prime\\prime}=0 has one of the following two parameter families as its general solution\n\\begin{aligned} y&=c_1e^x+c_2e^{-x}\\\\ y&=c_1\\cos(x)+c_2\\sin(x)\\\\ y&=c_1\\tan(x)+c_2\\sec(x)\\\\ y&=c_1+c_2x\\\\ \\end{aligned} Find the solution such that $y(0)=8$ and $y^\\prime(0)=6$: [ANS]", "answer_v1": [ "6*x+8" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The differential equation\ny^{\\prime\\prime}=0 has one of the following two parameter families as its general solution\n\\begin{aligned} y&=c_1e^x+c_2e^{-x}\\\\ y&=c_1\\cos(x)+c_2\\sin(x)\\\\ y&=c_1\\tan(x)+c_2\\sec(x)\\\\ y&=c_1+c_2x\\\\ \\end{aligned} Find the solution such that $y(0)=1$ and $y^\\prime(0)=10$: [ANS]", "answer_v2": [ "10*x+1" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The differential equation\ny^{\\prime\\prime}=0 has one of the following two parameter families as its general solution\n\\begin{aligned} y&=c_1e^x+c_2e^{-x}\\\\ y&=c_1\\cos(x)+c_2\\sin(x)\\\\ y&=c_1\\tan(x)+c_2\\sec(x)\\\\ y&=c_1+c_2x\\\\ \\end{aligned} Find the solution such that $y(0)=4$ and $y^\\prime(0)=7$: [ANS]", "answer_v3": [ "7*x+4" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0008", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "4", "keywords": [ "superposition" ], "problem_v1": "One of the following is a general solution of the homogeneous differential equation $8x^{2}y''+17xy'+y=0$\n\\begin{aligned} y&= \\frac{a}{x^2} + \\frac{b}{x^{\\frac{1} {64}}} \\\\ y&= \\frac{a}{x} + \\frac{b}{x^{{\\textstyle\\frac{1} {8}}}} \\\\ \\end{aligned} and one of the following is a solution to the nonhomogeneous equation $8x^{2}y''+17xy'+y=108x+6$ \\begin{aligned} y&=x^2-x \\\\ y&=6+6x \\\\ y&=e^x \\\\ \\end{aligned} By superposition, the general solution of the equation $8x^{2}y''+17xy'+y=108x+6$ is $y=$ [ANS]\nFind the solution with\n\\begin{aligned} y(1)&=7 \\\\ y^\\prime(1)&=4 \\\\ \\end{aligned} $y=$ [ANS]\nFortunately, since I already had it computed, my professor also asked me to find the Wronskian (using only the solutions to the homogeneous equation, without the coefficients a and b) [ANS]. The fundamental theorem shows that this solution is the unique solution to the IVP on the interval [ANS]", "answer_v1": [ "a/x+b/(x^0.125)+6+6*x", "3/x+(-8)/(x^0.125)+6+6*x", "7/(8*x^2.125)", "(0,infinity)" ], "answer_type_v1": [ "EX", "EX", "EX", "INT" ], "options_v1": [ [], [], [], [] ], "problem_v2": "One of the following is a general solution of the homogeneous differential equation $2x^{2}y''+5xy'+y=0$\n\\begin{aligned} y&= \\frac{a}{x^2} + \\frac{b}{x^{\\frac{1} {4}}} \\\\ y&= \\frac{a}{x} + \\frac{b}{x^{{\\textstyle\\frac{1} {2}}}} \\\\ \\end{aligned} and one of the following is a solution to the nonhomogeneous equation $2x^{2}y''+5xy'+y=12x+9$ \\begin{aligned} y&=x^2-x \\\\ y&=9+2x \\\\ y&=e^x \\\\ \\end{aligned} By superposition, the general solution of the equation $2x^{2}y''+5xy'+y=12x+9$ is $y=$ [ANS]\nFind the solution with\n\\begin{aligned} y(1)&=4 \\\\ y^\\prime(1)&=9 \\\\ \\end{aligned} $y=$ [ANS]\nFortunately, since I already had it computed, my professor also asked me to find the Wronskian (using only the solutions to the homogeneous equation, without the coefficients a and b) [ANS]. The fundamental theorem shows that this solution is the unique solution to the IVP on the interval [ANS]", "answer_v2": [ "a/x+b/(x^0.5)+9+2*x", "(-7)/x+0/(x^0.5)+9+2*x", "1/(2*x^2.5)", "(0,infinity)" ], "answer_type_v2": [ "EX", "EX", "EX", "INT" ], "options_v2": [ [], [], [], [] ], "problem_v3": "One of the following is a general solution of the homogeneous differential equation $4x^{2}y''+9xy'+y=0$\n\\begin{aligned} y&= \\frac{a}{x^2} + \\frac{b}{x^{\\frac{1} {16}}} \\\\ y&= \\frac{a}{x} + \\frac{b}{x^{{\\textstyle\\frac{1} {4}}}} \\\\ \\end{aligned} and one of the following is a solution to the nonhomogeneous equation $4x^{2}y''+9xy'+y=30x+6$ \\begin{aligned} y&=x^2-x \\\\ y&=6+3x \\\\ y&=e^x \\\\ \\end{aligned} By superposition, the general solution of the equation $4x^{2}y''+9xy'+y=30x+6$ is $y=$ [ANS]\nFind the solution with\n\\begin{aligned} y(1)&=6 \\\\ y^\\prime(1)&=3 \\\\ \\end{aligned} $y=$ [ANS]\nFortunately, since I already had it computed, my professor also asked me to find the Wronskian (using only the solutions to the homogeneous equation, without the coefficients a and b) [ANS]. The fundamental theorem shows that this solution is the unique solution to the IVP on the interval [ANS]", "answer_v3": [ "a/x+b/(x^0.25)+6+3*x", "1/x+(-4)/(x^0.25)+6+3*x", "3/(4*x^2.25)", "(0,infinity)" ], "answer_type_v3": [ "EX", "EX", "EX", "INT" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0009", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "2", "keywords": [ "calculus", "differential equations", "trigonometric functions" ], "problem_v1": "Check by differentiation that $y=4\\cos 3 t+5 \\sin 3 t$ is a solution to $y''+9 y=0$ by finding the terms in the sum: $y''=$ [ANS]\n$9 y=$ [ANS]\nSo $y''+9 y=$ [ANS]", "answer_v1": [ "-9*[4*cos(3*t)+5*sin(3*t)]", "9*[4*cos(3*t)+5*sin(3*t)]", "0" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Check by differentiation that $y=5\\cos t+3 \\sin t$ is a solution to $y''+y=0$ by finding the terms in the sum: $y''=$ [ANS]\n$y=$ [ANS]\nSo $y''+y=$ [ANS]", "answer_v2": [ "-1*[5*cos(1*t)+3*sin(1*t)]", "1*[5*cos(1*t)+3*sin(1*t)]", "0" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Check by differentiation that $y=4\\cos t+4 \\sin t$ is a solution to $y''+y=0$ by finding the terms in the sum: $y''=$ [ANS]\n$y=$ [ANS]\nSo $y''+y=$ [ANS]", "answer_v3": [ "-1*[4*cos(1*t)+4*sin(1*t)]", "1*[4*cos(1*t)+4*sin(1*t)]", "0" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0010", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "3", "keywords": [ "calculus", "integral", "differential equations", "higher derivatives" ], "problem_v1": "If $y=e^{5 t}$ is a solution to the differential equation \\frac{d^2y}{dt^2} -12 \\frac{dy}{dt} +ky=0, find the value of the constant $k$ and the general solution to this equation. $k=$ [ANS]\n$y=$ [ANS]\n(Use constants A, B, etc., for any constants in your solution formula.) (Use constants A, B, etc., for any constants in your solution formula.)", "answer_v1": [ "35", "A*e^(5*t)+B*e^(7*t)" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "If $y=e^{2 t}$ is a solution to the differential equation \\frac{d^2y}{dt^2} -7 \\frac{dy}{dt} +ky=0, find the value of the constant $k$ and the general solution to this equation. $k=$ [ANS]\n$y=$ [ANS]\n(Use constants A, B, etc., for any constants in your solution formula.) (Use constants A, B, etc., for any constants in your solution formula.)", "answer_v2": [ "10", "A*e^(2*t)+B*e^(5*t)" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "If $y=e^{3 t}$ is a solution to the differential equation \\frac{d^2y}{dt^2} -8 \\frac{dy}{dt} +ky=0, find the value of the constant $k$ and the general solution to this equation. $k=$ [ANS]\n$y=$ [ANS]\n(Use constants A, B, etc., for any constants in your solution formula.) (Use constants A, B, etc., for any constants in your solution formula.)", "answer_v3": [ "15", "A*e^(3*t)+B*e^(5*t)" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0011", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "3", "keywords": [ "differential equations" ], "problem_v1": "Suppose $x=e^{5t}$.\nFind the value of the expression $x'''-15x''+75x'-125x$ in terms of the variable $t$. (Enter the terms in the order given.) [ANS]+[ANS]+[ANS]+[ANS]\nSimplify your answer to the previous part to obtain a differential equation in terms of the dependent variable $x$ satisfied by $x=e^{5t}$. $x'''-15x''+75x'-125x=$ [ANS]\nIs $x=e^t$ a solution to your differential equation in the previous part? [ANS] Be sure you can justify your answer.", "answer_v1": [ "125*e^(5*t)", "(-375)*e^(5*t)", "375*e^(5*t)", "-125*e^(5*t)", "0", "No" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "NV", "TF" ], "options_v1": [ [], [], [], [], [], [ "Yes", "No" ] ], "problem_v2": "Suppose $x=e^{2t}$.\nFind the value of the expression $x'''-6x''+12x'-8x$ in terms of the variable $t$. (Enter the terms in the order given.) [ANS]+[ANS]+[ANS]+[ANS]\nSimplify your answer to the previous part to obtain a differential equation in terms of the dependent variable $x$ satisfied by $x=e^{2t}$. $x'''-6x''+12x'-8x=$ [ANS]\nIs $x=e^t$ a solution to your differential equation in the previous part? [ANS] Be sure you can justify your answer.", "answer_v2": [ "8*e^(2*t)", "(-24)*e^(2*t)", "24*e^(2*t)", "-8*e^(2*t)", "0", "No" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "NV", "TF" ], "options_v2": [ [], [], [], [], [], [ "Yes", "No" ] ], "problem_v3": "Suppose $x=e^{3t}$.\nFind the value of the expression $x'''-9x''+27x'-27x$ in terms of the variable $t$. (Enter the terms in the order given.) [ANS]+[ANS]+[ANS]+[ANS]\nSimplify your answer to the previous part to obtain a differential equation in terms of the dependent variable $x$ satisfied by $x=e^{3t}$. $x'''-9x''+27x'-27x=$ [ANS]\nIs $x=e^t$ a solution to your differential equation in the previous part? [ANS] Be sure you can justify your answer.", "answer_v3": [ "27*e^(3*t)", "(-81)*e^(3*t)", "81*e^(3*t)", "-27*e^(3*t)", "0", "No" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "NV", "TF" ], "options_v3": [ [], [], [], [], [], [ "Yes", "No" ] ] }, { "id": "Differential_equations_0012", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "1", "keywords": [ "First order differential equation" ], "problem_v1": "Suppose $y(t)=8 e^{-4 t}$ is a solution of the initial value problem $y^{\\,\\prime}+ky=0$, $y(0)=y_0$. What are the constants $k$ and $y_0$?\n$k=$ [ANS]\n$y_0=$ [ANS]", "answer_v1": [ "4", "8" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose $y(t)=2 e^{-5 t}$ is a solution of the initial value problem $y^{\\,\\prime}+ky=0$, $y(0)=y_0$. What are the constants $k$ and $y_0$?\n$k=$ [ANS]\n$y_0=$ [ANS]", "answer_v2": [ "5", "2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose $y(t)=4 e^{-4 t}$ is a solution of the initial value problem $y^{\\,\\prime}+ky=0$, $y(0)=y_0$. What are the constants $k$ and $y_0$?\n$k=$ [ANS]\n$y_0=$ [ANS]", "answer_v3": [ "4", "4" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0013", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "2", "keywords": [ "differential equations" ], "problem_v1": "Suppose $y=c_{1}e^{kx}+c_{2}e^{-kx}$ where $k > 0$ is a constant, and $c_1$ and $c_2$ are arbitrary constants. Find the following. Enter $c_1$ as c1 and $c_2$ as c2.\n$ \\frac{dy}{dx} =$ [ANS]\n$ \\frac{d^2y}{dx^2} =$ [ANS]\nRewrite your answer to the previous part in terms of $y$. $ \\frac{d^2y}{dx^2} =$ [ANS]\nFind the general solution to $ \\frac{d^2y}{dx^2} =64 y$. Enter your answer as an equation $y=\\ldots$. [ANS]", "answer_v1": [ "c1*e^(k*x)*k*ln(e)-c2*e^(-k*x)*k*ln(e)", "c1*e^(k*x)*k*ln(e)*k*ln(e)+c2*e^(-k*x)*k*ln(e)*k*ln(e)", "k^2*y", "y = c1*e^(8*x)+c2*e^(-8*x) or y = c1*e^(-8*x)+c2*e^[-(-8)*x]" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose $y=c_{1}e^{kx}+c_{2}e^{-kx}$ where $k > 0$ is a constant, and $c_1$ and $c_2$ are arbitrary constants. Find the following. Enter $c_1$ as c1 and $c_2$ as c2.\n$ \\frac{dy}{dx} =$ [ANS]\n$ \\frac{d^2y}{dx^2} =$ [ANS]\nRewrite your answer to the previous part in terms of $y$. $ \\frac{d^2y}{dx^2} =$ [ANS]\nFind the general solution to $ \\frac{d^2y}{dx^2} =9 y$. Enter your answer as an equation $y=\\ldots$. [ANS]", "answer_v2": [ "c1*e^(k*x)*k*ln(e)-c2*e^(-k*x)*k*ln(e)", "c1*e^(k*x)*k*ln(e)*k*ln(e)+c2*e^(-k*x)*k*ln(e)*k*ln(e)", "k^2*y", "y = c1*e^(3*x)+c2*e^(-3*x) or y = c1*e^(-3*x)+c2*e^[-(-3)*x]" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose $y=c_{1}e^{kx}+c_{2}e^{-kx}$ where $k > 0$ is a constant, and $c_1$ and $c_2$ are arbitrary constants. Find the following. Enter $c_1$ as c1 and $c_2$ as c2.\n$ \\frac{dy}{dx} =$ [ANS]\n$ \\frac{d^2y}{dx^2} =$ [ANS]\nRewrite your answer to the previous part in terms of $y$. $ \\frac{d^2y}{dx^2} =$ [ANS]\nFind the general solution to $ \\frac{d^2y}{dx^2} =25 y$. Enter your answer as an equation $y=\\ldots$. [ANS]", "answer_v3": [ "c1*e^(k*x)*k*ln(e)-c2*e^(-k*x)*k*ln(e)", "c1*e^(k*x)*k*ln(e)*k*ln(e)+c2*e^(-k*x)*k*ln(e)*k*ln(e)", "k^2*y", "y = c1*e^(5*x)+c2*e^(-5*x) or y = c1*e^(-5*x)+c2*e^[-(-5)*x]" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0014", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "2", "keywords": [ "differential equations" ], "problem_v1": "Find a non-constant solution to $(x^{\\,\\prime})^2+x^2=25$ using your knowledge of derivatives from basic calculus.\n$x(t)=$ [ANS]", "answer_v1": [ "5*cos(t)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find a non-constant solution to $(x^{\\,\\prime})^2+x^2=4$ using your knowledge of derivatives from basic calculus.\n$x(t)=$ [ANS]", "answer_v2": [ "2*cos(t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find a non-constant solution to $(x^{\\,\\prime})^2+x^2=9$ using your knowledge of derivatives from basic calculus.\n$x(t)=$ [ANS]", "answer_v3": [ "3*cos(t)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0015", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "2", "keywords": [ "differential equations" ], "problem_v1": "Let $y^{\\,\\prime\\prime}+2 y^{\\,\\prime}-48 y=0$.\nTry a solution of the form $y=e^{rx}$, for some unknown constant $r$, by substituting it into the differential equation. [ANS] $=0$.\nSimplify and factor your equation in the previous part as much as possible. Be sure to divide out any factors guaranteed to be nonzero. [ANS] $=0$.\nFind all values of $r$ such that $y=e^{rx}$ satisfies the differential equation. If there is more than one correct answer, enter your answers as a comma separated list. $r=$ [ANS]", "answer_v1": [ "r^2*e^(r*x)+2*r*e^(r*x)-48*e^(r*x)", "(r+8)*(r-6)", "(-8, 6)" ], "answer_type_v1": [ "EX", "EX", "UOL" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $y^{\\,\\prime\\prime}-7 y^{\\,\\prime}-18 y=0$.\nTry a solution of the form $y=e^{rx}$, for some unknown constant $r$, by substituting it into the differential equation. [ANS] $=0$.\nSimplify and factor your equation in the previous part as much as possible. Be sure to divide out any factors guaranteed to be nonzero. [ANS] $=0$.\nFind all values of $r$ such that $y=e^{rx}$ satisfies the differential equation. If there is more than one correct answer, enter your answers as a comma separated list. $r=$ [ANS]", "answer_v2": [ "r^2*e^(r*x)+(-7)*r*e^(r*x)-18*e^(r*x)", "(r+2)*(r-9)", "(-2, 9)" ], "answer_type_v2": [ "EX", "EX", "UOL" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $y^{\\,\\prime\\prime}-2 y^{\\,\\prime}-24 y=0$.\nTry a solution of the form $y=e^{rx}$, for some unknown constant $r$, by substituting it into the differential equation. [ANS] $=0$.\nSimplify and factor your equation in the previous part as much as possible. Be sure to divide out any factors guaranteed to be nonzero. [ANS] $=0$.\nFind all values of $r$ such that $y=e^{rx}$ satisfies the differential equation. If there is more than one correct answer, enter your answers as a comma separated list. $r=$ [ANS]", "answer_v3": [ "r^2*e^(r*x)+(-2)*r*e^(r*x)-24*e^(r*x)", "(r+4)*(r-6)", "(-4, 6)" ], "answer_type_v3": [ "EX", "EX", "UOL" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0016", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "1", "keywords": [ "differential equations" ], "problem_v1": "Find a solution to $ \\frac{dA}{dt} =7 A$ if $A(0)=6$.\n$A(t)=$ [ANS]", "answer_v1": [ "6*e^(7*t)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find a solution to $ \\frac{dA}{dt} =-10 A$ if $A(0)=3$.\n$A(t)=$ [ANS]", "answer_v2": [ "3*e^(-10*t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find a solution to $ \\frac{dA}{dt} =-7 A$ if $A(0)=4$.\n$A(t)=$ [ANS]", "answer_v3": [ "4*e^(-7*t)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0017", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "2", "keywords": [ "differential equations" ], "problem_v1": "Let $t^{2}y''+14 ty'+42 y=0$.\nFind all values of $r$ such that $y=t^{r}$ satisfies the differential equation for $t>0$. If there is more than one correct answer, enter your answers as a comma separated list.\n$r=$ [ANS]", "answer_v1": [ "(-7, -6)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Let $t^{2}y''+11 ty'+9 y=0$.\nFind all values of $r$ such that $y=t^{r}$ satisfies the differential equation for $t>0$. If there is more than one correct answer, enter your answers as a comma separated list.\n$r=$ [ANS]", "answer_v2": [ "(-1, -9)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Let $t^{2}y''+10 ty'+18 y=0$.\nFind all values of $r$ such that $y=t^{r}$ satisfies the differential equation for $t>0$. If there is more than one correct answer, enter your answers as a comma separated list.\n$r=$ [ANS]", "answer_v3": [ "(-3, -6)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0018", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "", "keywords": [ "differential equations" ], "problem_v1": "Let $y''+64 y=0$.\nFind all values of $r$ such that $y=c_{1}\\sin(rx)+c_{2}\\cos(rx)$ satisfies the differential equation. If there is more than one correct answer, enter your answers as a comma separated list.\n$r=$ [ANS]", "answer_v1": [ "(8, -8)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Let $y''+4 y=0$.\nFind all values of $r$ such that $y=c_{1}\\sin(rx)+c_{2}\\cos(rx)$ satisfies the differential equation. If there is more than one correct answer, enter your answers as a comma separated list.\n$r=$ [ANS]", "answer_v2": [ "(2, -2)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Let $y''+16 y=0$.\nFind all values of $r$ such that $y=c_{1}\\sin(rx)+c_{2}\\cos(rx)$ satisfies the differential equation. If there is more than one correct answer, enter your answers as a comma separated list.\n$r=$ [ANS]", "answer_v3": [ "(4, -4)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0019", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Verification of solutions", "level": "2", "keywords": [ "differential equations" ], "problem_v1": "Let $ y'= \\frac{xy}{x^2+25 y^2} $.\nFind all values of $r$ such that $x^2=ry^{2}\\ln y$ satisfies the differential equation. If there is more than one correct answer, enter your answers as a comma separated list.\n$r=$ [ANS]", "answer_v1": [ "50" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $ y'= \\frac{xy}{x^2+4 y^2} $.\nFind all values of $r$ such that $x^2=ry^{2}\\ln y$ satisfies the differential equation. If there is more than one correct answer, enter your answers as a comma separated list.\n$r=$ [ANS]", "answer_v2": [ "8" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $ y'= \\frac{xy}{x^2+9 y^2} $.\nFind all values of $r$ such that $x^2=ry^{2}\\ln y$ satisfies the differential equation. If there is more than one correct answer, enter your answers as a comma separated list.\n$r=$ [ANS]", "answer_v3": [ "18" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0020", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Classifications of differential equations", "level": "2", "keywords": [ "differential equation' 'match", "differential equations", "definitions" ], "problem_v1": "It can be helpful to classify a differential equation, so that we can predict the techniques that might help us to find a function which solves the equation. Two classifications are the order of the equation--(what is the highest number of derivatives involved) and whether or not the equation is linear. Linearity is important because the structure of the family of solutions to a linear equation is fairly simple. Linear equations can usually be solved completely and explicitly.\nDetermine whether or not each equation is linear:\n[ANS] 1. $ \\frac{d^2y}{dt^2} +\\sin(t+y)=\\sin t$ [ANS] 2. $ (1+y^2) \\frac{d^2y}{dt^2} +t \\frac{dy}{dt} +y=e^t$ [ANS] 3. $y''-y+y^2=0$ [ANS] 4. $y''-y+t^2=0$", "answer_v1": [ "2NONLINEAR", "2NONLINEAR", "2NONLINEAR", "2Linear" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "1Linear", "2Linear", "3Linear", "4Linear", "5Linear", "1Nonlinear", "2Nonlinear", "3Nonlinear", "4Nonlinear", "5Nonlinear" ], [ "1Linear", "2Linear", "3Linear", "4Linear", "5Linear", "1Nonlinear", "2Nonlinear", "3Nonlinear", "4Nonlinear", "5Nonlinear" ], [ "1Linear", "2Linear", "3Linear", "4Linear", "5Linear", "1Nonlinear", "2Nonlinear", "3Nonlinear", "4Nonlinear", "5Nonlinear" ], [ "1Linear", "2Linear", "3Linear", "4Linear", "5Linear", "1Nonlinear", "2Nonlinear", "3Nonlinear", "4Nonlinear", "5Nonlinear" ] ], "problem_v2": "It can be helpful to classify a differential equation, so that we can predict the techniques that might help us to find a function which solves the equation. Two classifications are the order of the equation--(what is the highest number of derivatives involved) and whether or not the equation is linear. Linearity is important because the structure of the family of solutions to a linear equation is fairly simple. Linear equations can usually be solved completely and explicitly.\nDetermine whether or not each equation is linear:\n[ANS] 1. $ (1+y^2) \\frac{d^2y}{dt^2} +t \\frac{dy}{dt} +y=e^t$ [ANS] 2. $y''-y+t^2=0$ [ANS] 3. $ \\frac{d^2y}{dt^2} +\\sin(t+y)=\\sin t$ [ANS] 4. $ t^2 \\frac{d^2y}{dt^2} +t \\frac{dy}{dt} +2y=\\sin t$", "answer_v2": [ "2NONLINEAR", "2LINEAR", "2NONLINEAR", "2Linear" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "1Linear", "2Linear", "3Linear", "4Linear", "5Linear", "1Nonlinear", "2Nonlinear", "3Nonlinear", "4Nonlinear", "5Nonlinear" ], [ "1Linear", "2Linear", "3Linear", "4Linear", "5Linear", "1Nonlinear", "2Nonlinear", "3Nonlinear", "4Nonlinear", "5Nonlinear" ], [ "1Linear", "2Linear", "3Linear", "4Linear", "5Linear", "1Nonlinear", "2Nonlinear", "3Nonlinear", "4Nonlinear", "5Nonlinear" ], [ "1Linear", "2Linear", "3Linear", "4Linear", "5Linear", "1Nonlinear", "2Nonlinear", "3Nonlinear", "4Nonlinear", "5Nonlinear" ] ], "problem_v3": "It can be helpful to classify a differential equation, so that we can predict the techniques that might help us to find a function which solves the equation. Two classifications are the order of the equation--(what is the highest number of derivatives involved) and whether or not the equation is linear. Linearity is important because the structure of the family of solutions to a linear equation is fairly simple. Linear equations can usually be solved completely and explicitly.\nDetermine whether or not each equation is linear:\n[ANS] 1. $ \\frac{d^4y}{dt^4} + \\frac{d^3y}{dt^3} + \\frac{d^2y}{dt^2} + \\frac{dy}{dt} =1$ [ANS] 2. $ t^2 \\frac{d^2y}{dt^2} +t \\frac{dy}{dt} +2y=\\sin t$ [ANS] 3. $ \\frac{d^2y}{dt^2} +\\sin(t+y)=\\sin t$ [ANS] 4. $ \\frac{d^3y}{dt^3} +t \\frac{dy}{dt} +(cos^2(t))y=t^3$", "answer_v3": [ "4LINEAR", "2LINEAR", "2NONLINEAR", "3linear" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "1Linear", "2Linear", "3Linear", "4Linear", "5Linear", "1Nonlinear", "2Nonlinear", "3Nonlinear", "4Nonlinear", "5Nonlinear" ], [ "1Linear", "2Linear", "3Linear", "4Linear", "5Linear", "1Nonlinear", "2Nonlinear", "3Nonlinear", "4Nonlinear", "5Nonlinear" ], [ "1Linear", "2Linear", "3Linear", "4Linear", "5Linear", "1Nonlinear", "2Nonlinear", "3Nonlinear", "4Nonlinear", "5Nonlinear" ], [ "1Linear", "2Linear", "3Linear", "4Linear", "5Linear", "1Nonlinear", "2Nonlinear", "3Nonlinear", "4Nonlinear", "5Nonlinear" ] ] }, { "id": "Differential_equations_0021", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Classifications of differential equations", "level": "2", "keywords": [ "linear' 'differential' 'equation" ], "problem_v1": "For the differential equation $ \\frac{dy}{dx} =y+e^x$ check all that apply [ANS] A. separable B. logistic C. nonhomogeneous D. linear E. autonomous F. homogeneous", "answer_v1": [ "CD" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "For the differential equation $ \\frac{dy}{dx} =y+e^x$ check all that apply [ANS] A. logistic B. autonomous C. separable D. linear E. homogeneous F. nonhomogeneous", "answer_v2": [ "DF" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "For the differential equation $ \\frac{dy}{dx} =y+e^x$ check all that apply [ANS] A. autonomous B. logistic C. linear D. separable E. nonhomogeneous F. homogeneous", "answer_v3": [ "CE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Differential_equations_0022", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Classifications of differential equations", "level": "2", "keywords": [ "linear' 'differential' 'equation" ], "problem_v1": "Which of the following are first order linear differential equations? [ANS] A. $ \\frac{dy}{dx} =y^2-3y$ B. $\\left( \\frac{dy}{dx} \\right)^2+\\cos(x)y=5$ C. $x \\frac{dy}{dx} -4y=x^6e^x$ D. $\\sin(x) \\frac{dy}{dx} -3y=0$ E. $ \\frac{dP}{dt} +2tP=P+4t-2$ F. $ \\frac{d^2y}{dx^2} +\\sin(x) \\frac{dy}{dx} =\\cos(x)$", "answer_v1": [ "CDE" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Which of the following are first order linear differential equations? [ANS] A. $ \\frac{d^2y}{dx^2} +\\sin(x) \\frac{dy}{dx} =\\cos(x)$ B. $\\left( \\frac{dy}{dx} \\right)^2+\\cos(x)y=5$ C. $ \\frac{dy}{dx} =y^2-3y$ D. $x \\frac{dy}{dx} -4y=x^6e^x$ E. $\\sin(x) \\frac{dy}{dx} -3y=0$ F. $ \\frac{dP}{dt} +2tP=P+4t-2$", "answer_v2": [ "DEF" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Which of the following are first order linear differential equations? [ANS] A. $ \\frac{d^2y}{dx^2} +\\sin(x) \\frac{dy}{dx} =\\cos(x)$ B. $x \\frac{dy}{dx} -4y=x^6e^x$ C. $\\sin(x) \\frac{dy}{dx} -3y=0$ D. $ \\frac{dy}{dx} =y^2-3y$ E. $ \\frac{dP}{dt} +2tP=P+4t-2$ F. $\\left( \\frac{dy}{dx} \\right)^2+\\cos(x)y=5$", "answer_v3": [ "BCE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Differential_equations_0023", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Classifications of differential equations", "level": "2", "keywords": [ "differential equations", "existence", "uniqueness" ], "problem_v1": "Put the differential equation $ 8 t y+e^t y^{\\,\\prime}= \\frac{y}{t^2+64} $ into the form $y^{\\,\\prime}+p(t) y=g(t)$ and find $p(t)$ and $g(t)$.\n$p(t)=$ [ANS]\n$g(t)=$ [ANS]\nIs the differential equation $ 8 t y+e^t y^{\\,\\prime}= \\frac{y}{t^2+64} $ linear and homogeneous, linear and nonhomogeneous, or nonlinear?\nAnswer: [ANS]", "answer_v1": [ "8*t*e^(-t)-e^(-t)/(t^2+64)", "0", "linear and homogeneous" ], "answer_type_v1": [ "EX", "NV", "MCS" ], "options_v1": [ [], [], [ "linear and homogeneous", "linear and nonhomogeneous", "nonlinear" ] ], "problem_v2": "Put the differential equation $ 2 t y+e^t y^{\\,\\prime}= \\frac{y}{t^2+4} $ into the form $y^{\\,\\prime}+p(t) y=g(t)$ and find $p(t)$ and $g(t)$.\n$p(t)=$ [ANS]\n$g(t)=$ [ANS]\nIs the differential equation $ 2 t y+e^t y^{\\,\\prime}= \\frac{y}{t^2+4} $ linear and homogeneous, linear and nonhomogeneous, or nonlinear?\nAnswer: [ANS]", "answer_v2": [ "2*t*e^(-t)-e^(-t)/(t^2+4)", "0", "linear and homogeneous" ], "answer_type_v2": [ "EX", "NV", "MCS" ], "options_v2": [ [], [], [ "linear and homogeneous", "linear and nonhomogeneous", "nonlinear" ] ], "problem_v3": "Put the differential equation $ 4 t y+e^t y^{\\,\\prime}= \\frac{y}{t^2+16} $ into the form $y^{\\,\\prime}+p(t) y=g(t)$ and find $p(t)$ and $g(t)$.\n$p(t)=$ [ANS]\n$g(t)=$ [ANS]\nIs the differential equation $ 4 t y+e^t y^{\\,\\prime}= \\frac{y}{t^2+16} $ linear and homogeneous, linear and nonhomogeneous, or nonlinear?\nAnswer: [ANS]", "answer_v3": [ "4*t*e^(-t)-e^(-t)/(t^2+16)", "0", "linear and homogeneous" ], "answer_type_v3": [ "EX", "NV", "MCS" ], "options_v3": [ [], [], [ "linear and homogeneous", "linear and nonhomogeneous", "nonlinear" ] ] }, { "id": "Differential_equations_0024", "subject": "Differential_equations", "topic": "Introductory concepts", "subtopic": "Classifications of differential equations", "level": "2", "keywords": [ "differential equations" ], "problem_v1": "Determine the order of the given differential equation and state whether the equation is linear or nonlinear. \\ \\frac{d^{8}y}{dt^{8} }+\\sin(t+y)=\\sin(6 t)\n(a) The order of this differential equation is [ANS].\n(b) The equation is [ANS].", "answer_v1": [ "8", "Nonlinear" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "Linear", "Nonlinear" ] ], "problem_v2": "Determine the order of the given differential equation and state whether the equation is linear or nonlinear. \\ \\frac{d^{2}y}{dt^{2} }+\\sin(t+y)=\\sin(9 t)\n(a) The order of this differential equation is [ANS].\n(b) The equation is [ANS].", "answer_v2": [ "2", "Nonlinear" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "Linear", "Nonlinear" ] ], "problem_v3": "Determine the order of the given differential equation and state whether the equation is linear or nonlinear. \\ \\frac{d^{4}y}{dt^{4} }+\\sin(t+y)=\\sin(6 t)\n(a) The order of this differential equation is [ANS].\n(b) The equation is [ANS].", "answer_v3": [ "4", "Nonlinear" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "Linear", "Nonlinear" ] ] }, { "id": "Differential_equations_0025", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Linear", "level": "3", "keywords": [ "differential equations", "first order", "integrating factor" ], "problem_v1": "Find a solution to the initial value problem\ny^{\\,\\prime}+\\sin(t) y=g(t), \\ \\ \\ y(0)=8, that is continuous on the interval $\\lbrack 0, 2\\pi \\rbrack$ where\ng(t)=\\left\\lbrace \\begin{array}{rcl} \\sin(t) && \\mathrm{if} \\ 0 \\leq t \\leq \\pi, \\\\-\\sin(t) && \\mathrm{if} \\ \\pi < t \\leq 2\\pi. \\end{array} \\right.\n$\\begin{array}{ccc}\\hline y(t)= \\Bigg\\lbrace & & [ANS] \\mbox{if} 0 \\leq t \\leq \\pi, [ANS] \\mbox{if} \\pi < t \\leq 2\\pi. \\\\ \\hline \\end{array}$", "answer_v1": [ "1+7*e^[cos(t)-1]", "-1+7*e^[cos(t)-1]+2*e^[cos(t)+1]" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Find a solution to the initial value problem\ny^{\\,\\prime}+\\sin(t) y=g(t), \\ \\ \\ y(0)=3, that is continuous on the interval $\\lbrack 0, 2\\pi \\rbrack$ where\ng(t)=\\left\\lbrace \\begin{array}{rcl} \\sin(t) && \\mathrm{if} \\ 0 \\leq t \\leq \\pi, \\\\-\\sin(t) && \\mathrm{if} \\ \\pi < t \\leq 2\\pi. \\end{array} \\right.\n$\\begin{array}{ccc}\\hline y(t)= \\Bigg\\lbrace & & [ANS] \\mbox{if} 0 \\leq t \\leq \\pi, [ANS] \\mbox{if} \\pi < t \\leq 2\\pi. \\\\ \\hline \\end{array}$", "answer_v2": [ "1+2*e^[cos(t)-1]", "-1+2*e^[cos(t)-1]+2*e^[cos(t)+1]" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Find a solution to the initial value problem\ny^{\\,\\prime}+\\sin(t) y=g(t), \\ \\ \\ y(0)=5, that is continuous on the interval $\\lbrack 0, 2\\pi \\rbrack$ where\ng(t)=\\left\\lbrace \\begin{array}{rcl} \\sin(t) && \\mathrm{if} \\ 0 \\leq t \\leq \\pi, \\\\-\\sin(t) && \\mathrm{if} \\ \\pi < t \\leq 2\\pi. \\end{array} \\right.\n$\\begin{array}{ccc}\\hline y(t)= \\Bigg\\lbrace & & [ANS] \\mbox{if} 0 \\leq t \\leq \\pi, [ANS] \\mbox{if} \\pi < t \\leq 2\\pi. \\\\ \\hline \\end{array}$", "answer_v3": [ "1+4*e^[cos(t)-1]", "-1+4*e^[cos(t)-1]+2*e^[cos(t)+1]" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0026", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Exact", "level": "4", "keywords": [ "exact equation", "differential equation" ], "problem_v1": "Solve $y^{3}-\\left(14x+6\\right)+3xy^{2} y'=0$. (Denote the arbitrary constant in your solution by $C$.) $y$=[ANS]", "answer_v1": [ "[(7*x^2+6*x+C)/x]^0.333333" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve $y^{3}-\\left(2x+8\\right)+3xy^{2} y'=0$. (Denote the arbitrary constant in your solution by $C$.) $y$=[ANS]", "answer_v2": [ "[(1*x^2+8*x+C)/x]^0.333333" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve $y^{3}-\\left(6x+6\\right)+3xy^{2} y'=0$. (Denote the arbitrary constant in your solution by $C$.) $y$=[ANS]", "answer_v3": [ "[(3*x^2+6*x+C)/x]^0.333333" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0027", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Exact", "level": "3", "keywords": [ "differential equations", "separable", "differential equation' 'linear" ], "problem_v1": "Solve the initial value problem\n6 \\big(\\sin(t) \\frac{dy}{dt} +\\cos(t) y \\big)=\\cos(t)\\sin^{6}(t), for $0 < t < \\pi$ and $y(\\pi/2)=13.$ $y=$ [ANS].", "answer_v1": [ "12.9762*[sin(t)]^(-1)+0.0238095*[sin(t)]^6" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the initial value problem\n9 \\big(\\sin(t) \\frac{dy}{dt} +\\cos(t) y \\big)=\\cos(t)\\sin^{2}(t), for $0 < t < \\pi$ and $y(\\pi/2)=4.$ $y=$ [ANS].", "answer_v2": [ "3.96296*[sin(t)]^(-1)+0.037037*[sin(t)]^2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the initial value problem\n6 \\big(\\sin(t) \\frac{dy}{dt} +\\cos(t) y \\big)=\\cos(t)\\sin^{3}(t), for $0 < t < \\pi$ and $y(\\pi/2)=7.$ $y=$ [ANS].", "answer_v3": [ "6.95833*[sin(t)]^(-1)+0.0416667*[sin(t)]^3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0028", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Exact", "level": "3", "keywords": [], "problem_v1": "The differential equation y+y^5=\\left(y^4+4x\\right)y' can be written in differential form: M(x,y) \\, dx+N(x,y) \\, dy=0 where $M(x,y)=$ [ANS], and $N(x,y)=$ [ANS]. The term $M(x,y) \\, dx+N(x,y) \\, dy$ becomes an exact differential if the left hand side above is divided by $y^5$. Integrating that new equation, the solution of the differential equation is [ANS] $=C$.", "answer_v1": [ "y+1*y^5", "-y^4-4*x", "-[ln(y)]+x/(y^4)+1*x" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "The differential equation y+4 y^3=\\left(y^4+2x\\right)y' can be written in differential form: M(x,y) \\, dx+N(x,y) \\, dy=0 where $M(x,y)=$ [ANS], and $N(x,y)=$ [ANS]. The term $M(x,y) \\, dx+N(x,y) \\, dy$ becomes an exact differential if the left hand side above is divided by $y^3$. Integrating that new equation, the solution of the differential equation is [ANS] $=C$.", "answer_v2": [ "y+4*y^3", "-y^4-2*x", "-y^2/2+x/(y^2)+4*x" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "The differential equation y+y^7=\\left(y^4+6x\\right)y' can be written in differential form: M(x,y) \\, dx+N(x,y) \\, dy=0 where $M(x,y)=$ [ANS], and $N(x,y)=$ [ANS]. The term $M(x,y) \\, dx+N(x,y) \\, dy$ becomes an exact differential if the left hand side above is divided by $y^7$. Integrating that new equation, the solution of the differential equation is [ANS] $=C$.", "answer_v3": [ "y+1*y^7", "-y^4-6*x", "-y^(-2)/-2+x/(y^6)+1*x" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0029", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Exact", "level": "3", "keywords": [ "differential equations" ], "problem_v1": "Solve the following differential equation:\n$\\ (8xy^{2}-3)dx+(8x^{2}y+4)dy=0$. [ANS]=constant.", "answer_v1": [ "8*x^2*y^2-3*x+4*y" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the following differential equation:\n$\\ (2xy^{2}-3)dx+(2x^{2}y+4)dy=0$. [ANS]=constant.", "answer_v2": [ "2*x^2*y^2-3*x+4*y" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the following differential equation:\n$\\ (4xy^{2}-3)dx+(4x^{2}y+4)dy=0$. [ANS]=constant.", "answer_v3": [ "4*x^2*y^2-3*x+4*y" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0030", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Exact", "level": "3", "keywords": [ "Differential equaltion" ], "problem_v1": "Solve the following initial-value problem:\n$ (x+y)^{2}dx+(2xy+x^{2}-1)dy=0, \\ y(1)=5$ [ANS]=0.", "answer_v1": [ "-76+x^3+3*x^2*y+3*x*y^2-3*y" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the following initial-value problem:\n$ (x+y)^{2}dx+(2xy+x^{2}-1)dy=0, \\ y(1)=2$ [ANS]=0.", "answer_v2": [ "-13+x^3+3*x^2*y+3*x*y^2-3*y" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the following initial-value problem:\n$ (x+y)^{2}dx+(2xy+x^{2}-1)dy=0, \\ y(1)=3$ [ANS]=0.", "answer_v3": [ "-28+x^3+3*x^2*y+3*x*y^2-3*y" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0031", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Exact", "level": "3", "keywords": [ "differential equations" ], "problem_v1": "Solve the following initial value problem:\n$ (y^{2}\\cos(x)-3x^{2}y-2x)dx+(2y\\sin(x)-x^{3}+\\ln(y))dy=0,\\ y(0)=e^{4}$. [ANS]=0.", "answer_v1": [ "y^2*sin(x)-x^3*y-x^2+y*ln(y)-y-[e^4*ln(e^4)-e^4]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the following initial value problem:\n$ (y^{2}\\cos(x)-3x^{2}y-2x)dx+(2y\\sin(x)-x^{3}+\\ln(y))dy=0,\\ y(0)=e^{1}$. [ANS]=0.", "answer_v2": [ "y^2*sin(x)-x^3*y-x^2+y*ln(y)-y-[e*ln(e)-e]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the following initial value problem:\n$ (y^{2}\\cos(x)-3x^{2}y-2x)dx+(2y\\sin(x)-x^{3}+\\ln(y))dy=0,\\ y(0)=e^{2}$. [ANS]=0.", "answer_v3": [ "y^2*sin(x)-x^3*y-x^2+y*ln(y)-y-[e^2*ln(e^2)-e^2]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0032", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "3", "keywords": [ "calculus", "differential equations", "logistic equation", "population growth", "exponential growth", "s-curve", "sigmoid" ], "problem_v1": "Consider the logistic equation \\dot{y}=y\\left(1-y\\right)\n(a) Find the solution satisfying $y_1(0)=14$ and $y_2(0)=-3$. $y_1(t)=$ [ANS]\n$y_2(t)=$ [ANS]\n(b) Find the time $t$ when $y_1(t)=7$. $t=$ [ANS]\n(c) When does $y_2(t)$ become infinite? $t$=[ANS]", "answer_v1": [ "1/[1-e^(-t)/(14/13)]", "1/[1-e^(-t)/(3/4)]", "0.0800427", "0.287682" ], "answer_type_v1": [ "EX", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the logistic equation \\dot{y}=y\\left(1-y\\right)\n(a) Find the solution satisfying $y_1(0)=6$ and $y_2(0)=-1$. $y_1(t)=$ [ANS]\n$y_2(t)=$ [ANS]\n(b) Find the time $t$ when $y_1(t)=3$. $t=$ [ANS]\n(c) When does $y_2(t)$ become infinite? $t$=[ANS]", "answer_v2": [ "1/[1-e^(-t)/(6/5)]", "1/[1-e^(-t)/(1/2)]", "0.223144", "0.693147" ], "answer_type_v2": [ "EX", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the logistic equation \\dot{y}=y\\left(1-y\\right)\n(a) Find the solution satisfying $y_1(0)=8$ and $y_2(0)=-2$. $y_1(t)=$ [ANS]\n$y_2(t)=$ [ANS]\n(b) Find the time $t$ when $y_1(t)=4$. $t=$ [ANS]\n(c) When does $y_2(t)$ become infinite? $t$=[ANS]", "answer_v3": [ "1/[1-e^(-t)/(8/7)]", "1/[1-e^(-t)/(2/3)]", "0.154151", "0.405465" ], "answer_type_v3": [ "EX", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0033", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "3", "keywords": [ "calculus", "differential equations", "separation of variables" ], "problem_v1": "Solve $(t^2+25)\\, \\frac{dx}{dt} =(x^2+16)$, using separation of variables, given the inital condition $x(0)=4$. $x=$ [ANS]", "answer_v1": [ "4*tan(0.8*atan(t/5)+0.785398)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve $(t^2+4)\\, \\frac{dx}{dt} =(x^2+36)$, using separation of variables, given the inital condition $x(0)=6$. $x=$ [ANS]", "answer_v2": [ "6*tan(3*atan(t/2)+0.785398)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve $(t^2+9)\\, \\frac{dx}{dt} =(x^2+25)$, using separation of variables, given the inital condition $x(0)=5$. $x=$ [ANS]", "answer_v3": [ "5*tan(1.66667*atan(t/3)+0.785398)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0034", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "3", "keywords": [ "calculus", "differential equations", "separation of variables" ], "problem_v1": "Solve the initial value problem $ \\frac{dy}{dx} =(x-6)(y-9),\\, y(0)=7$. $y=$ [ANS]", "answer_v1": [ "9-2*e^(0.5*x^2-6*x)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the initial value problem $ \\frac{dy}{dx} =(x-2)(y-3),\\, y(0)=10$. $y=$ [ANS]", "answer_v2": [ "7*e^(0.5*x^2-2*x)+3" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the initial value problem $ \\frac{dy}{dx} =(x-3)(y-5),\\, y(0)=7$. $y=$ [ANS]", "answer_v3": [ "2*e^(0.5*x^2-3*x)+5" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0035", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "3", "keywords": [ "calculus", "differential equations", "separation of variables" ], "problem_v1": "Solve $y'=4x^{4}y^2$, using separation of variables, given the inital condition $y(0)=13$. $y=$ [ANS]", "answer_v1": [ "-1/(4/5*x^5-1/13)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve $y'=x^{6}y^2$, using separation of variables, given the inital condition $y(0)=3$. $y=$ [ANS]", "answer_v2": [ "-1/(1/7*x^7-1/3)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve $y'=2x^{5}y^2$, using separation of variables, given the inital condition $y(0)=6$. $y=$ [ANS]", "answer_v3": [ "-1/(2/6*x^6-1/6)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0036", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "3", "keywords": [ "calculus", "differential equations", "separation of variables" ], "problem_v1": "Solve \\frac{dy}{dt} =y\\tan{t}, using separation of variables given the inital condition $y(0)=16$. Assume the function is defined for $-\\pi/20$, and use the initial condition $x(1)=4$. $x=$ [ANS]", "answer_v1": [ "4^(t^(7))" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the differential equation \\frac{dx}{dt} = \\frac{x\\ln x}{t} Assume $x, t >0$, and use the initial condition $x(1)=5$. $x=$ [ANS]", "answer_v2": [ "5^(t^(1))" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the differential equation \\frac{dx}{dt} = \\frac{3x\\ln x}{t} Assume $x, t >0$, and use the initial condition $x(1)=4$. $x=$ [ANS]", "answer_v3": [ "4^(t^(3))" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0059", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "4", "keywords": [ "calculus", "integral", "differential equations", "separable", "solution of differential equations" ], "problem_v1": "Solve the differential equation {dR\\over dx}=a (R^2+16). Assume $a$ is a non-zero constant, and use $C$ for any constant of integration that you may have in your answer. $R=$ [ANS]", "answer_v1": [ "4*tan(4*(a*x+C))" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the differential equation {dR\\over dx}=a (R^2+1). Assume $a$ is a non-zero constant, and use $C$ for any constant of integration that you may have in your answer. $R=$ [ANS]", "answer_v2": [ "1*tan(1*(a*x+C))" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the differential equation {dR\\over dx}=a (R^2+4). Assume $a$ is a non-zero constant, and use $C$ for any constant of integration that you may have in your answer. $R=$ [ANS]", "answer_v3": [ "2*tan(2*(a*x+C))" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0060", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "4", "keywords": [ "calculus", "integral", "differential equations", "separable", "solution of differential equations" ], "problem_v1": "Find the solution to the differential equation \\frac{dz}{dt} =8 t e^{5 z} that passes through the origin. $z=$ [ANS]", "answer_v1": [ "(-1/5)*ln(1 - 8*5*t^2/2)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the solution to the differential equation \\frac{dz}{dt} =2 t e^{7 z} that passes through the origin. $z=$ [ANS]", "answer_v2": [ "(-1/7)*ln(1 - 2*7*t^2/2)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the solution to the differential equation \\frac{dz}{dt} =4 t e^{5 z} that passes through the origin. $z=$ [ANS]", "answer_v3": [ "(-1/5)*ln(1 - 4*5*t^2/2)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0062", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "3", "keywords": [ "calculus", "integral", "differential equations" ], "problem_v1": "Let $A$ and $k$ be positive constants.\nWhich of the given functions is a solution to $ \\frac{dy}{dt} =k(A y-1)$? [ANS] A. $y=-A+C e^{kt}$ B. $y=A^{-1}+C e^{-Akt}$ C. $y=A^{-1}+C e^{Akt}$ D. $y=A+C e^{-kt}$ E. $y=A+C e^{kt}$ F. $y=-A+C e^{-kt}$", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Let $A$ and $k$ be positive constants.\nWhich of the given functions is a solution to $ \\frac{dy}{dt} =k(y-A)$? [ANS] A. $y=A+C e^{kt}$ B. $y=A^{-1}+C e^{-Akt}$ C. $y=A^{-1}+C e^{Akt}$ D. $y=-A+C e^{kt}$ E. $y=A+C e^{-kt}$ F. $y=-A+C e^{-kt}$", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Let $A$ and $k$ be positive constants.\nWhich of the given functions is a solution to $ \\frac{dy}{dt} =k(y+A)$? [ANS] A. $y=A^{-1}+C e^{Akt}$ B. $y=A^{-1}+C e^{-Akt}$ C. $y=A+C e^{kt}$ D. $y=-A+C e^{kt}$ E. $y=A+C e^{-kt}$ F. $y=-A+C e^{-kt}$", "answer_v3": [ "D" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Differential_equations_0063", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "2", "keywords": [ "differential equation", "calculus", "antiderivatives'\"" ], "problem_v1": "Find the solution of the initial value problem. ${dP\\over dt}=20 e^t, \\quad P(0)=25$ $P=$ [ANS]", "answer_v1": [ "20*e^t+5" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the solution of the initial value problem. ${dP\\over dt}=5 e^t, \\quad P(0)=15$ $P=$ [ANS]", "answer_v2": [ "5*e^t+10" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the solution of the initial value problem. ${dP\\over dt}=10 e^t, \\quad P(0)=15$ $P=$ [ANS]", "answer_v3": [ "10*e^t+5" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0064", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "2", "keywords": [ "Differantial Equations", "Separation of Variables", "General Solution", "Specific Solution" ], "problem_v1": "Using separation of variables, solve the differential equation, (6+x^{10}) \\frac{dy}{dx} = \\frac{x^{9}}{y} . Use C to represent the arbitrary constant. $\\small{y^2}$=[ANS]", "answer_v1": [ "0.2*ln(6+x^10)+C" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Using separation of variables, solve the differential equation, (10+x^{2}) \\frac{dy}{dx} = \\frac{x^{1}}{y} . Use C to represent the arbitrary constant. $\\small{y^2}$=[ANS]", "answer_v2": [ "ln(10+x^2)+C" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Using separation of variables, solve the differential equation, (7+x^{4}) \\frac{dy}{dx} = \\frac{x^{3}}{y} . Use C to represent the arbitrary constant. $\\small{y^2}$=[ANS]", "answer_v3": [ "0.5*ln(7+x^4)+C" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0065", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "5", "keywords": [ "Differential Equation", "Initial Condition", "Trigonometry", "differential equation' 'linear", "differential equations", "separable" ], "problem_v1": "Find the particular solution of the differential equation \\frac{dy}{dx} +y\\cos(x)=5\\cos(x) satisfying the initial condition $y(0)=7$. Answer: $y$=[ANS]\nYour answer should be a function of $x$.", "answer_v1": [ "5 + 2*e^(-sin(x))" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the particular solution of the differential equation \\frac{dy}{dx} +y\\cos(x)=2\\cos(x) satisfying the initial condition $y(0)=4$. Answer: $y$=[ANS]\nYour answer should be a function of $x$.", "answer_v2": [ "2 + 2*e^(-sin(x))" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the particular solution of the differential equation \\frac{dy}{dx} +y\\cos(x)=3\\cos(x) satisfying the initial condition $y(0)=5$. Answer: $y$=[ANS]\nYour answer should be a function of $x$.", "answer_v3": [ "3 + 2*e^(-sin(x))" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0066", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "3", "keywords": [ "differential equations", "first order", "integrals as solutions" ], "problem_v1": "Set up an integral for solving $ \\frac{dy}{dx} =\\sin(x^{6})+x$ when $y(4)=14$.\n$\\begin{array}{ccccc}\\hline y(x)=[ANS]+\\int & & t=[ANS] t=[ANS] & & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "14", "x", "4", "[sin(t^6)+t]*dt" ], "answer_type_v1": [ "NV", "EX", "NV", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Set up an integral for solving $ \\frac{dy}{dx} =\\sin(x^{3})+x$ when $y(1)=19$.\n$\\begin{array}{ccccc}\\hline y(x)=[ANS]+\\int & & t=[ANS] t=[ANS] & & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "19", "x", "1", "[sin(t^3)+t]*dt" ], "answer_type_v2": [ "NV", "EX", "NV", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Set up an integral for solving $ \\frac{dy}{dx} =\\sin(x^{4})+x$ when $y(2)=15$.\n$\\begin{array}{ccccc}\\hline y(x)=[ANS]+\\int & & t=[ANS] t=[ANS] & & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "15", "x", "2", "[sin(t^4)+t]*dt" ], "answer_type_v3": [ "NV", "EX", "NV", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0067", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "3", "keywords": [ "differential equations", "first order", "integrals as solutions" ], "problem_v1": "Solve $y^{\\,\\prime}=y^3$ if $y(0)=4$.\n$y(x)=$ [ANS]", "answer_v1": [ "[2*(1/32-x)]^(-1/2)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve $y^{\\,\\prime}=y^3$ if $y(0)=1$.\n$y(x)=$ [ANS]", "answer_v2": [ "[2*(1/2-x)]^(-1/2)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve $y^{\\,\\prime}=y^3$ if $y(0)=2$.\n$y(x)=$ [ANS]", "answer_v3": [ "[2*(1/8-x)]^(-1/2)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0068", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "4", "keywords": [ "differential equations", "first order", "separable differential equations" ], "problem_v1": "Consider the initial value problem\ny^{\\,\\prime}=5 y^2, \\ \\ \\ y(0)=y_0. For what value(s) of $y_0$ will the solution have a vertical asymptote at $t=6$ and a $t$-interval of existence $-\\infty < t < 6$?\n$y_0=$ [ANS]", "answer_v1": [ "1/30" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Consider the initial value problem\ny^{\\,\\prime}=2 y^2, \\ \\ \\ y(0)=y_0. For what value(s) of $y_0$ will the solution have a vertical asymptote at $t=8$ and a $t$-interval of existence $-\\infty < t < 8$?\n$y_0=$ [ANS]", "answer_v2": [ "1/16" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Consider the initial value problem\ny^{\\,\\prime}=3 y^2, \\ \\ \\ y(0)=y_0. For what value(s) of $y_0$ will the solution have a vertical asymptote at $t=6$ and a $t$-interval of existence $-\\infty < t < 6$?\n$y_0=$ [ANS]", "answer_v3": [ "1/18" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0069", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "3", "keywords": [ "differential equations", "first order", "separable differential equations" ], "problem_v1": "Find the equation of the solution to $ \\frac{dy}{dx} =x^{6} y$ through the point $(x,y)=(1,5)$. [ANS]", "answer_v1": [ "y = 5/[e^(1/7)]*e^(x^7/7)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the equation of the solution to $ \\frac{dy}{dx} =x^{8} y$ through the point $(x,y)=(1,2)$. [ANS]", "answer_v2": [ "y = 2/[e^(1/9)]*e^(x^9/9)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the equation of the solution to $ \\frac{dy}{dx} =x^{6} y$ through the point $(x,y)=(1,3)$. [ANS]", "answer_v3": [ "y = 3/[e^(1/7)]*e^(x^7/7)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0070", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "4", "keywords": [ "differential equations", "first order", "separable differential equations" ], "problem_v1": "Solve the differential equation $ \\frac{dy}{dx} = \\frac{x}{36 y} $.\nFind an implicit solution and put your answer in the following form: [ANS]=constant.\nFind the equation of the solution through the point $(x,y)=(6,1)$. [ANS]\nFind the equation of the solution through the point $(x,y)=(0,-4)$. Your answer should be of the form $y=f(x)$. [ANS]", "answer_v1": [ "y^2-(x/6)^2", "6*y-x = 0", "y = -[sqrt((x/6)^2+16)]" ], "answer_type_v1": [ "EX", "EQ", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Solve the differential equation $ \\frac{dy}{dx} = \\frac{x}{4 y} $.\nFind an implicit solution and put your answer in the following form: [ANS]=constant.\nFind the equation of the solution through the point $(x,y)=(2,1)$. [ANS]\nFind the equation of the solution through the point $(x,y)=(0,-7)$. Your answer should be of the form $y=f(x)$. [ANS]", "answer_v2": [ "y^2-(x/2)^2", "2*y-x = 0", "y = -[sqrt((x/2)^2+49)]" ], "answer_type_v2": [ "EX", "EQ", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Solve the differential equation $ \\frac{dy}{dx} = \\frac{x}{9 y} $.\nFind an implicit solution and put your answer in the following form: [ANS]=constant.\nFind the equation of the solution through the point $(x,y)=(3,1)$. [ANS]\nFind the equation of the solution through the point $(x,y)=(0,-6)$. Your answer should be of the form $y=f(x)$. [ANS]", "answer_v3": [ "y^2-(x/3)^2", "3*y-x = 0", "y = -[sqrt((x/3)^2+36)]" ], "answer_type_v3": [ "EX", "EQ", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0071", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "3", "keywords": [ "Differential Equation", "Separable" ], "problem_v1": "Solve the separable differential equation.\ny'=8 y^2 Use the following initial condition: $y(8)=7$ $y=$ [ANS]\nNote: Your answer should be a function of $x$.", "answer_v1": [ "-1/(8*x-1/7-8^2)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the separable differential equation.\ny'=2 y^2 Use the following initial condition: $y(2)=11$ $y=$ [ANS]\nNote: Your answer should be a function of $x$.", "answer_v2": [ "-1/(2*x-1/11-2^2)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the separable differential equation.\ny'=4 y^2 Use the following initial condition: $y(4)=8$ $y=$ [ANS]\nNote: Your answer should be a function of $x$.", "answer_v3": [ "-1/(4*x-1/8-4^2)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0072", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "3", "keywords": [ "Differential equation", "Linear 1st order" ], "problem_v1": "A. Let g(t) be the solution of the initial value problem\n6 t \\frac{dy}{dt} +y=0, t > 0, with $g(1)=1.$ Find $g(t)$. $g(t)=$ [ANS]", "answer_v1": [ "t**(-1/6 )" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "A. Let g(t) be the solution of the initial value problem\n2 t \\frac{dy}{dt} +y=0, t > 0, with $g(1)=1.$ Find $g(t)$. $g(t)=$ [ANS]", "answer_v2": [ "t**(-1/2 )" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "A. Let g(t) be the solution of the initial value problem\n3 t \\frac{dy}{dt} +y=0, t > 0, with $g(1)=1.$ Find $g(t)$. $g(t)=$ [ANS]", "answer_v3": [ "t**(-1/3 )" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0073", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Separable", "level": "3", "keywords": [ "differential equations" ], "problem_v1": "Find the general solution of the differential equation y'=e^{8x}-6x. (Don't forget+C.)\n$y=$ [ANS]", "answer_v1": [ "0.125*e^(8*x)-3*x^2+C" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the general solution of the differential equation y'=e^{4x}-9x. (Don't forget+C.)\n$y=$ [ANS]", "answer_v2": [ "0.25*e^(4*x)-4.5*x^2+C" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the general solution of the differential equation y'=e^{5x}-6x. (Don't forget+C.)\n$y=$ [ANS]", "answer_v3": [ "0.2*e^(5*x)-3*x^2+C" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0074", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Substitutions", "level": "3", "keywords": [ "differential", "equation", "bernoulli", "linear", "substitution" ], "problem_v1": "A Bernoulli differential equation is one of the form \\frac{dy}{dx} +P(x)y=Q(x)y^n. Observe that, if $n=0$ or $1$, the Bernoulli equation is linear. For other values of $n$, the substitution $u=y^{1-n}$ transforms the Bernoulli equation into the linear equation \\frac{du}{dx} +(1-n)P(x)u=(1-n)Q(x).\nUse an appropriate substitution to solve the equation y'- \\frac{8}{x} y= \\frac{y^4}{x^{13} }, and find the solution that satisfies $y(1)=1.$\n$y(x)=$ [ANS].", "answer_v1": [ "( (- 3)/(12*x^{12}) + 1.25/x^{24} )^(- 1/3)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "A Bernoulli differential equation is one of the form \\frac{dy}{dx} +P(x)y=Q(x)y^n. Observe that, if $n=0$ or $1$, the Bernoulli equation is linear. For other values of $n$, the substitution $u=y^{1-n}$ transforms the Bernoulli equation into the linear equation \\frac{du}{dx} +(1-n)P(x)u=(1-n)Q(x).\nUse an appropriate substitution to solve the equation y'- \\frac{2}{x} y= \\frac{y^5}{x^{2} }, and find the solution that satisfies $y(1)=1.$\n$y(x)=$ [ANS].", "answer_v2": [ "( (- 4)/(7*x^{1}) + 1.57142857142857/x^{8} )^(- 1/4)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "A Bernoulli differential equation is one of the form \\frac{dy}{dx} +P(x)y=Q(x)y^n. Observe that, if $n=0$ or $1$, the Bernoulli equation is linear. For other values of $n$, the substitution $u=y^{1-n}$ transforms the Bernoulli equation into the linear equation \\frac{du}{dx} +(1-n)P(x)u=(1-n)Q(x).\nUse an appropriate substitution to solve the equation y'- \\frac{4}{x} y= \\frac{y^4}{x^{4} }, and find the solution that satisfies $y(1)=1.$\n$y(x)=$ [ANS].", "answer_v3": [ "( (- 3)/(9*x^{3}) + 1.33333333333333/x^{12} )^(- 1/3)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0075", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Substitutions", "level": "3", "keywords": [ "differential equation' 'linear" ], "problem_v1": "GUESS one function $y(t)$ which solves the problem below, by determining the general form the function might take and then evaluating some coefficients.\n \\frac{dy}{dt} +6 y=\\exp(3 t) Find $y(t)$. $y(t)=$ [ANS].", "answer_v1": [ " (exp(3* t ))/(9)+c e^(-6*t)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "GUESS one function $y(t)$ which solves the problem below, by determining the general form the function might take and then evaluating some coefficients.\n \\frac{dy}{dt} +2 y=\\exp(5 t) Find $y(t)$. $y(t)=$ [ANS].", "answer_v2": [ " (exp(5* t ))/(7)+c e^(-2*t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "GUESS one function $y(t)$ which solves the problem below, by determining the general form the function might take and then evaluating some coefficients.\n \\frac{dy}{dt} +3 y=\\exp(4 t) Find $y(t)$. $y(t)=$ [ANS].", "answer_v3": [ " (exp(4* t ))/(7)+c e^{-3*t}" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0076", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Substitutions", "level": "4", "keywords": [ "differential equations", "first order", "separable differential equations" ], "problem_v1": "Solve the initial value problem $ y^{\\,\\prime}=(x+y-4)^2$ with $y(0)=0$.\nTo solve this, we should use the substitution $u=$ [ANS]\n$u^{\\,\\prime}=$ [ANS]\nEnter derivatives using prime notation (e.g., you would enter $y^{\\,\\prime}$ for $ \\frac{dy}{dx} $).\nAfter the substitution from the previous part, we obtain the following differential equation in $x, u, u^{\\,\\prime}$. [ANS]\nThe solution to the original initial value problem is described by the following equation in $x, y$. [ANS]", "answer_v1": [ "x+y-4", "1+y", "u", "y = tan(x+atan(-4))-x+4" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Solve the initial value problem $ y^{\\,\\prime}=(x+y-1)^2$ with $y(0)=0$.\nTo solve this, we should use the substitution $u=$ [ANS]\n$u^{\\,\\prime}=$ [ANS]\nEnter derivatives using prime notation (e.g., you would enter $y^{\\,\\prime}$ for $ \\frac{dy}{dx} $).\nAfter the substitution from the previous part, we obtain the following differential equation in $x, u, u^{\\,\\prime}$. [ANS]\nThe solution to the original initial value problem is described by the following equation in $x, y$. [ANS]", "answer_v2": [ "x+y-1", "1+y", "u", "y = tan(x+atan(-1))-x+1" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Solve the initial value problem $ y^{\\,\\prime}=(x+y-2)^2$ with $y(0)=0$.\nTo solve this, we should use the substitution $u=$ [ANS]\n$u^{\\,\\prime}=$ [ANS]\nEnter derivatives using prime notation (e.g., you would enter $y^{\\,\\prime}$ for $ \\frac{dy}{dx} $).\nAfter the substitution from the previous part, we obtain the following differential equation in $x, u, u^{\\,\\prime}$. [ANS]\nThe solution to the original initial value problem is described by the following equation in $x, y$. [ANS]", "answer_v3": [ "x+y-2", "1+y", "u", "y = tan(x+atan(-2))-x+2" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0077", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Substitutions", "level": "4", "keywords": [ "differential equations", "first order", "separable differential equations" ], "problem_v1": "Sometimes a change of variable can be used to convert a differential equation $y^{\\,\\prime}=f(t,y)$ into a separable equation. One common change of variable technique is as follows.\nConsider a differential equation of the form $y^{\\,\\prime}=f(\\alpha t+\\beta y+\\gamma)$, where $\\alpha, \\beta$, and $\\gamma$ are constants. Use the change of variable $z=\\alpha t+\\beta y+\\gamma$ to rewrite the differential equation as a separable equation of the form $z^{\\,\\prime}=g(z)$.\nSolve the initial value problem\ny^{\\,\\prime}=(t+y)^{2}-1, \\ \\ \\ y(3)=5.\n(a) $g(z)=$ [ANS]\n(b) $y(t)=$ [ANS]", "answer_v1": [ "z^2", "-1/(t-3-1/8)-t" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Sometimes a change of variable can be used to convert a differential equation $y^{\\,\\prime}=f(t,y)$ into a separable equation. One common change of variable technique is as follows.\nConsider a differential equation of the form $y^{\\,\\prime}=f(\\alpha t+\\beta y+\\gamma)$, where $\\alpha, \\beta$, and $\\gamma$ are constants. Use the change of variable $z=\\alpha t+\\beta y+\\gamma$ to rewrite the differential equation as a separable equation of the form $z^{\\,\\prime}=g(z)$.\nSolve the initial value problem\ny^{\\,\\prime}=(t+y)^{2}-1, \\ \\ \\ y(1)=6.\n(a) $g(z)=$ [ANS]\n(b) $y(t)=$ [ANS]", "answer_v2": [ "z^2", "-1/(t-1-1/7)-t" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Sometimes a change of variable can be used to convert a differential equation $y^{\\,\\prime}=f(t,y)$ into a separable equation. One common change of variable technique is as follows.\nConsider a differential equation of the form $y^{\\,\\prime}=f(\\alpha t+\\beta y+\\gamma)$, where $\\alpha, \\beta$, and $\\gamma$ are constants. Use the change of variable $z=\\alpha t+\\beta y+\\gamma$ to rewrite the differential equation as a separable equation of the form $z^{\\,\\prime}=g(z)$.\nSolve the initial value problem\ny^{\\,\\prime}=(t+y)^{2}-1, \\ \\ \\ y(1)=5.\n(a) $g(z)=$ [ANS]\n(b) $y(t)=$ [ANS]", "answer_v3": [ "z^2", "-1/(t-1-1/6)-t" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0078", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Equilibrium points and phase lines", "level": "2", "keywords": [ "calculus", "differential equations", "logistic equation", "population growth", "exponential growth", "s-curve", "sigmoid" ], "problem_v1": "Let $y(t)$ be a solution of $\\dot{y}= \\frac{1}{7} y (1- \\frac{y}{7} )$ such that $y(0)=14$. Determine $ \\lim_{t \\to \\infty} y(t)$ without finding $y(t)$ explicitly. $ \\lim_{t \\to \\infty} y(t)$=[ANS]", "answer_v1": [ "7" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $y(t)$ be a solution of $\\dot{y}= \\frac{1}{2} y (1- \\frac{y}{2} )$ such that $y(0)=4$. Determine $ \\lim_{t \\to \\infty} y(t)$ without finding $y(t)$ explicitly. $ \\lim_{t \\to \\infty} y(t)$=[ANS]", "answer_v2": [ "2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $y(t)$ be a solution of $\\dot{y}= \\frac{1}{4} y (1- \\frac{y}{4} )$ such that $y(0)=8$. Determine $ \\lim_{t \\to \\infty} y(t)$ without finding $y(t)$ explicitly. $ \\lim_{t \\to \\infty} y(t)$=[ANS]", "answer_v3": [ "4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0079", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Equilibrium points and phase lines", "level": "4", "keywords": [ "differential equation' 'stability" ], "problem_v1": "Given the differential equation $x'=-(x+2.5)*(x+1)^3(x-0.5)^2(x-2.5)$. List the constant (or equilibrium) solutions to this differential equation in increasing order and indicate whether or not these equilibria are stable, semi-stable, or unstable. (It helps to) [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS]", "answer_v1": [ "-2.5", "STABLE", "-1", "unstable", "0.5", "semi-stable", "2.5", "stable" ], "answer_type_v1": [ "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS" ], "options_v1": [ [], [ "stable", "unstable", "semi-stable" ], [], [ "stable", "unstable", "semi-stable" ], [], [ "stable", "unstable", "semi-stable" ], [], [ "stable", "unstable", "semi-stable" ] ], "problem_v2": "Given the differential equation $x'=(x+3.5)*(x+1.5)^3(x+0.5)^2(x-1)$. List the constant (or equilibrium) solutions to this differential equation in increasing order and indicate whether or not these equilibria are stable, semi-stable, or unstable. (It helps to) [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS]", "answer_v2": [ "-3.5", "UNSTABLE", "-1.5", "stable", "-0.5", "semi-stable", "1", "unstable" ], "answer_type_v2": [ "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS" ], "options_v2": [ [], [ "stable", "unstable", "semi-stable" ], [], [ "stable", "unstable", "semi-stable" ], [], [ "stable", "unstable", "semi-stable" ], [], [ "stable", "unstable", "semi-stable" ] ], "problem_v3": "Given the differential equation $x'=-(x+3.5)*(x+2)^3(x+1)^2(x-0.5)$. List the constant (or equilibrium) solutions to this differential equation in increasing order and indicate whether or not these equilibria are stable, semi-stable, or unstable. (It helps to) [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS] [ANS]", "answer_v3": [ "-3.5", "STABLE", "-2", "unstable", "-1", "semi-stable", "0.5", "stable" ], "answer_type_v3": [ "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS" ], "options_v3": [ [], [ "stable", "unstable", "semi-stable" ], [], [ "stable", "unstable", "semi-stable" ], [], [ "stable", "unstable", "semi-stable" ], [], [ "stable", "unstable", "semi-stable" ] ] }, { "id": "Differential_equations_0080", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Equilibrium points and phase lines", "level": "4", "keywords": [ "differential", "equation", "constant", "solution" ], "problem_v1": "A function $y(t)$ satisfies the differential equation \\frac{dy}{dt} =-y^4+2 y^3+15 y^2.\n(a) What are the constant solutions of this equation? Separate your answers by commas. [ANS].\n(b) For what values of $y$ is $y$ increasing? [ANS] $< y <$ [ANS].", "answer_v1": [ "(-3, 0, 5)", "-3", "5" ], "answer_type_v1": [ "UOL", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A function $y(t)$ satisfies the differential equation \\frac{dy}{dt} =-y^4-2 y^3+63 y^2.\n(a) What are the constant solutions of this equation? Separate your answers by commas. [ANS].\n(b) For what values of $y$ is $y$ increasing? [ANS] $< y <$ [ANS].", "answer_v2": [ "(-9, 0, 7)", "-9", "7" ], "answer_type_v2": [ "UOL", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A function $y(t)$ satisfies the differential equation \\frac{dy}{dt} =-y^4-2 y^3+35 y^2.\n(a) What are the constant solutions of this equation? Separate your answers by commas. [ANS].\n(b) For what values of $y$ is $y$ increasing? [ANS] $< y <$ [ANS].", "answer_v3": [ "(-7, 0, 5)", "-7", "5" ], "answer_type_v3": [ "UOL", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0081", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Equilibrium points and phase lines", "level": "3", "keywords": [ "differential equations", "first order", "slope fields", "direction fields" ], "problem_v1": "Find an autonomous differential equation with equilibrium solutions at $y=n/5$ for $n=0, \\pm 1, \\pm 2, \\ldots$\n$ \\frac{dy}{dt} =$ [ANS]", "answer_v1": [ "sin(5*pi*y)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find an autonomous differential equation with equilibrium solutions at $y=n/2$ for $n=0, \\pm 1, \\pm 2, \\ldots$\n$ \\frac{dy}{dt} =$ [ANS]", "answer_v2": [ "sin(2*pi*y)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find an autonomous differential equation with equilibrium solutions at $y=n/3$ for $n=0, \\pm 1, \\pm 2, \\ldots$\n$ \\frac{dy}{dt} =$ [ANS]", "answer_v3": [ "sin(3*pi*y)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0082", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Equilibrium points and phase lines", "level": "3", "keywords": [ "differential equations", "first order", "slope fields", "direction fields" ], "problem_v1": "Find an autonomous differential equation with all of the following properties:\nequilibrium solutions at $y=0$ and $y=8$,\n$y^{\\,\\prime} > 0$ for $0 < y < 8$, and\n$y^{\\,\\prime} < 0$ for $-\\infty < y < 0$ and $8 < y < \\infty$. $ \\frac{dy}{dt} =$ [ANS]", "answer_v1": [ "y*(8-y)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find an autonomous differential equation with all of the following properties:\nequilibrium solutions at $y=0$ and $y=3$,\n$y^{\\,\\prime} > 0$ for $0 < y < 3$, and\n$y^{\\,\\prime} < 0$ for $-\\infty < y < 0$ and $3 < y < \\infty$. $ \\frac{dy}{dt} =$ [ANS]", "answer_v2": [ "y*(3-y)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find an autonomous differential equation with all of the following properties:\nequilibrium solutions at $y=0$ and $y=5$,\n$y^{\\,\\prime} > 0$ for $0 < y < 5$, and\n$y^{\\,\\prime} < 0$ for $-\\infty < y < 0$ and $5 < y < \\infty$. $ \\frac{dy}{dt} =$ [ANS]", "answer_v3": [ "y*(5-y)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0088", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "calculus", "differential equations", "exponential growth" ], "problem_v1": "A continuous annuity with withdrawal rate N=$\\$1{,}700$/year and interest rate r=4\\% is funded by an initial deposit $P_0$.\n(a) When will the annuity run out of funds if $P_0=\\$38{,}500$? The annuity runs out after approximately [ANS] years. Answer to the nearest whole year. Answer to the nearest whole year. (b) Which initial deposit $P_0$ yields a constant balance? $P_0$=\\$ [ANS]", "answer_v1": [ "59", "42500" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A continuous annuity with withdrawal rate N=$\\$600$/year and interest rate r=5\\% is funded by an initial deposit $P_0$.\n(a) When will the annuity run out of funds if $P_0=\\$11{,}000$? The annuity runs out after approximately [ANS] years. Answer to the nearest whole year. Answer to the nearest whole year. (b) Which initial deposit $P_0$ yields a constant balance? $P_0$=\\$ [ANS]", "answer_v2": [ "50", "12000" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A continuous annuity with withdrawal rate N=$\\$1{,}000$/year and interest rate r=4\\% is funded by an initial deposit $P_0$.\n(a) When will the annuity run out of funds if $P_0=\\$23{,}000$? The annuity runs out after approximately [ANS] years. Answer to the nearest whole year. Answer to the nearest whole year. (b) Which initial deposit $P_0$ yields a constant balance? $P_0$=\\$ [ANS]", "answer_v3": [ "63", "25000" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0089", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "calculus", "differential equations", "exponential growth" ], "problem_v1": "An initial deposit of $\\$7{,}500$ is placed in a bank account. What is the minimum interest rate that the bank must pay to allow continuous withdrawals at a rate of $\\$900$/year to continue indefinitely? minimum interest rate=[ANS] \\%", "answer_v1": [ "12" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "An initial deposit of $\\$5{,}000$ is placed in a bank account. What is the minimum interest rate that the bank must pay to allow continuous withdrawals at a rate of $\\$1{,}000$/year to continue indefinitely? minimum interest rate=[ANS] \\%", "answer_v2": [ "20" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "An initial deposit of $\\$6{,}000$ is placed in a bank account. What is the minimum interest rate that the bank must pay to allow continuous withdrawals at a rate of $\\$900$/year to continue indefinitely? minimum interest rate=[ANS] \\%", "answer_v3": [ "15" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0090", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "calculus", "differential equations", "exponential growth" ], "problem_v1": "Find the minimum initial deposit that will allow an annuity to pay out $\\$1{,}700$/year indefinitely if it earns interest at a rate of 4\\%. minimum initial deposit=\\$ [ANS]", "answer_v1": [ "42500" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the minimum initial deposit that will allow an annuity to pay out $\\$600$/year indefinitely if it earns interest at a rate of 5\\%. minimum initial deposit=\\$ [ANS]", "answer_v2": [ "12000" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the minimum initial deposit that will allow an annuity to pay out $\\$1{,}000$/year indefinitely if it earns interest at a rate of 4\\%. minimum initial deposit=\\$ [ANS]", "answer_v3": [ "25000" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0091", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "Algebra", "Exponential", "Logarithmic", "Applications", "calculus", "differential equation", "exponential growth" ], "problem_v1": "The count in a bateria culture was 700 after 15 minutes and 1700 after 35 minutes. What was the initial size of the culture? [ANS] Find the doubling period. [ANS] Find the population after 75 minutes. [ANS] When will the population reach 11000. [ANS]", "answer_v1": [ "359.819651832671", "15.6236827381026", "10026.5306122449", "77.0885900615809" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The count in a bateria culture was 100 after 20 minutes and 1300 after 30 minutes. What was the initial size of the culture? [ANS] Find the doubling period. [ANS] Find the population after 120 minutes. [ANS] When will the population reach 11000. [ANS]", "answer_v2": [ "0.591715976331361", "2.7023815442732", "13785849184900", "38.3258213349068" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The count in a bateria culture was 300 after 15 minutes and 1600 after 30 minutes. What was the initial size of the culture? [ANS] Find the doubling period. [ANS] Find the population after 70 minutes. [ANS] When will the population reach 12000. [ANS]", "answer_v3": [ "56.25", "6.21108368067956", "138925.912222176", "48.054940739905" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0092", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "differential equation' 'application" ], "problem_v1": "A young person with no initial capital invests $k$ dollars per year in a retirement account at an annual rate of return $0.09$. Assume that investments are made continuously and that the return is compounded continuously.\nDetermine a formula for the sum $S(t)$--(this will involve the parameter $k$): $S(t)=$ [ANS]\nWhat value of $k$ will provide $3446000$ dollars in $47$ years? $k=$ [ANS]", "answer_v1": [ "(k/0.09)*(exp(0.09*t) - 1 )", "4579.92729537138" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A young person with no initial capital invests $k$ dollars per year in a retirement account at an annual rate of return $0.03$. Assume that investments are made continuously and that the return is compounded continuously.\nDetermine a formula for the sum $S(t)$--(this will involve the parameter $k$): $S(t)=$ [ANS]\nWhat value of $k$ will provide $1513000$ dollars in $43$ years? $k=$ [ANS]", "answer_v2": [ "(k/0.03)*(exp(0.03*t) - 1 )", "17240.2885835238" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A young person with no initial capital invests $k$ dollars per year in a retirement account at an annual rate of return $0.05$. Assume that investments are made continuously and that the return is compounded continuously.\nDetermine a formula for the sum $S(t)$--(this will involve the parameter $k$): $S(t)=$ [ANS]\nWhat value of $k$ will provide $2042000$ dollars in $46$ years? $k=$ [ANS]", "answer_v3": [ "(k/0.05)*(exp(0.05*t) - 1 )", "11377.0809223098" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0093", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "differential equation' 'application' 'exponential growth", "Differential Equation", "population growth", "calculus", "exponential growth", "derivatives" ], "problem_v1": "A bacteria culture starts with $780$ bacteria and grows at a rate proportional to its size. After $4$ hours there will be $3120$ bacteria.\n(a) Express the population after $t$ hours as a function of $t$. population: [ANS] (function of t)\n(b) What will be the population after $6$ hours? [ANS]\n(c) How long will it take for the population to reach $2450$? [ANS]", "answer_v1": [ "780*(2.71828182845905^(0.346573590279973*t))", "6240", "3.30247144028064" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A bacteria culture starts with $160$ bacteria and grows at a rate proportional to its size. After $6$ hours there will be $960$ bacteria.\n(a) Express the population after $t$ hours as a function of $t$. population: [ANS] (function of t)\n(b) What will be the population after $3$ hours? [ANS]\n(c) How long will it take for the population to reach $1670$? [ANS]", "answer_v2": [ "160*(2.71828182845905^(0.298626578204676*t))", "391.918358845309", "7.85397302637093" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A bacteria culture starts with $380$ bacteria and grows at a rate proportional to its size. After $5$ hours there will be $1900$ bacteria.\n(a) Express the population after $t$ hours as a function of $t$. population: [ANS] (function of t)\n(b) What will be the population after $4$ hours? [ANS]\n(c) How long will it take for the population to reach $2090$? [ANS]", "answer_v3": [ "380*(2.71828182845905^(0.32188758248682*t))", "1377.08136098762", "5.29609772165793" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0094", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "differential equation' 'linear", "Differential Equation", "population growth" ], "problem_v1": "You have $900$ dollars in your bank account. Suppose your money is compounded every month at a rate of $0.3$ percent per month.\n(a) How much do you have after $t$ years? $y(t)=$ [ANS] (function of $t$)\n(b) How much do you have after $100$ months? $y(100)=$ [ANS]", "answer_v1": [ "900*(1+0.3/100)**(12*t)", "1214.32744742984" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "You have $500$ dollars in your bank account. Suppose your money is compounded every month at a rate of $0.5$ percent per month.\n(a) How much do you have after $t$ years? $y(t)=$ [ANS] (function of $t$)\n(b) How much do you have after $70$ months? $y(70)=$ [ANS]", "answer_v2": [ "500*(1+0.5/100)**(12*t)", "708.915263726959" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "You have $600$ dollars in your bank account. Suppose your money is compounded every month at a rate of $0.4$ percent per month.\n(a) How much do you have after $t$ years? $y(t)=$ [ANS] (function of $t$)\n(b) How much do you have after $70$ months? $y(70)=$ [ANS]", "answer_v3": [ "600*(1+0.4/100)**(12*t)", "793.434621560575" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0095", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "differential equation' 'application", "Differential Equation", "population growth" ], "problem_v1": "An unknown radioactive element decays into non-radioactive substances. In $780$ days the radioactivity of a sample decreases by $46$ percent.\n(a) What is the half-life of the element? half-life: [ANS] (days)\n(b) How long will it take for a sample of $100$ mg to decay to $77$ mg? time needed: [ANS] (days)", "answer_v1": [ "877.421230770157", "330.84891558169" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "An unknown radioactive element decays into non-radioactive substances. In $160$ days the radioactivity of a sample decreases by $27$ percent.\n(a) What is the half-life of the element? half-life: [ANS] (days)\n(b) How long will it take for a sample of $100$ mg to decay to $49$ mg? time needed: [ANS] (days)", "answer_v2": [ "352.398355340808", "362.669479615417" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "An unknown radioactive element decays into non-radioactive substances. In $380$ days the radioactivity of a sample decreases by $45$ percent.\n(a) What is the half-life of the element? half-life: [ANS] (days)\n(b) How long will it take for a sample of $100$ mg to decay to $57$ mg? time needed: [ANS] (days)", "answer_v3": [ "440.581510143846", "357.296702325826" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0096", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "differential", "equation" ], "problem_v1": "How long will it take an investment to triple in value if the interest rate is 6\\% compounded continuously? Answer: [ANS] years.", "answer_v1": [ "18.3102048111352" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "How long will it take an investment to double in value if the interest rate is 8\\% compounded continuously? Answer: [ANS] years.", "answer_v2": [ "8.66433975699932" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "How long will it take an investment to double in value if the interest rate is 6\\% compounded continuously? Answer: [ANS] years.", "answer_v3": [ "11.5524530093324" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0097", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "2", "keywords": [], "problem_v1": "The population of rabbits on an island increases by $0.8 \\%$ per year. Write out an equation in Leibniz notation that models this situation: [ANS]=[ANS]\nwhere $R(t)$ is the number of rabbits on the island and t is in years. Solve the equation if at time zero the rabbit population is $790$ rabbits. $R(t)$=[ANS]", "answer_v1": [ "dR/dt", "0.008*R", "790*e^(0.008*t)" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "The population of rabbits on an island increases by $0.3 \\%$ per year. Write out an equation in Leibniz notation that models this situation: [ANS]=[ANS]\nwhere $R(t)$ is the number of rabbits on the island and t is in years. Solve the equation if at time zero the rabbit population is $965$ rabbits. $R(t)$=[ANS]", "answer_v2": [ "dR/dt", "0.003*R", "965*e^(0.003*t)" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "The population of rabbits on an island increases by $0.5 \\%$ per year. Write out an equation in Leibniz notation that models this situation: [ANS]=[ANS]\nwhere $R(t)$ is the number of rabbits on the island and t is in years. Solve the equation if at time zero the rabbit population is $802$ rabbits. $R(t)$=[ANS]", "answer_v3": [ "dR/dt", "0.005*R", "802*e^(0.005*t)" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0098", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "2", "keywords": [], "problem_v1": "Uranium-238 decays at a rate of $1.55414\\times 10^{-10}$ percent per year. If u(t) is the amount of uranium in grams at time t in years, write an equation that models this statement in Leibniz notation: [ANS]=[ANS]\nSolve the equation if at time zero there are $87$ grams of uranium. $u(t)$=[ANS]", "answer_v1": [ "du/dt", "(1.55414E-12)*u", "87*e^[(1.55414E-12)*t]" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Uranium-238 decays at a rate of $1.55414\\times 10^{-10}$ percent per year. If u(t) is the amount of uranium in grams at time t in years, write an equation that models this statement in Leibniz notation: [ANS]=[ANS]\nSolve the equation if at time zero there are $54$ grams of uranium. $u(t)$=[ANS]", "answer_v2": [ "du/dt", "(1.55414E-12)*u", "54*e^[(1.55414E-12)*t]" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Uranium-238 decays at a rate of $1.55414\\times 10^{-10}$ percent per year. If u(t) is the amount of uranium in grams at time t in years, write an equation that models this statement in Leibniz notation: [ANS]=[ANS]\nSolve the equation if at time zero there are $65$ grams of uranium. $u(t)$=[ANS]", "answer_v3": [ "du/dt", "(1.55414E-12)*u", "65*e^[(1.55414E-12)*t]" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0099", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [], "problem_v1": "The knowledge that a civilization can amass in a year is proportional to the knowledge it currently posses. If K(t) is the knowledge that a certain civilization possesses at time t, this civilization can add 10\\% to its knowledge each year, and this civilization possesses $80000$ volumes of knowledge at time 0, write an initial volume problem for the amount of knowledge the civilization possesses: [ANS] $=0$ $K(0)=$ [ANS]\nNote: use K,K' instead of K(t), K'(t) in your answers. After you set up your differential equation you will have to set it equal to zero so that WeBWorK will understand your answer, do this in a way so that the highest order derivative has a positive coefficient.", "answer_v1": [ "K", "80000" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The knowledge that a civilization can amass in a year is proportional to the knowledge it currently posses. If K(t) is the knowledge that a certain civilization possesses at time t, this civilization can add 10\\% to its knowledge each year, and this civilization possesses $15000$ volumes of knowledge at time 0, write an initial volume problem for the amount of knowledge the civilization possesses: [ANS] $=0$ $K(0)=$ [ANS]\nNote: use K,K' instead of K(t), K'(t) in your answers. After you set up your differential equation you will have to set it equal to zero so that WeBWorK will understand your answer, do this in a way so that the highest order derivative has a positive coefficient.", "answer_v2": [ "K", "15000" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The knowledge that a civilization can amass in a year is proportional to the knowledge it currently posses. If K(t) is the knowledge that a certain civilization possesses at time t, this civilization can add 10\\% to its knowledge each year, and this civilization possesses $35000$ volumes of knowledge at time 0, write an initial volume problem for the amount of knowledge the civilization possesses: [ANS] $=0$ $K(0)=$ [ANS]\nNote: use K,K' instead of K(t), K'(t) in your answers. After you set up your differential equation you will have to set it equal to zero so that WeBWorK will understand your answer, do this in a way so that the highest order derivative has a positive coefficient.", "answer_v3": [ "K", "35000" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0100", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "calculus", "integral", "differential equations", "modeling", "population growth" ], "problem_v1": "Assuming that Switzerland's population is growing exponentially at a continuous rate of 0.23 percent a year and that the 1988 population was 6.7 million, write an expression for the population as a function of time in years. (Let $t=0$ in 1988.) $P=$ [ANS]", "answer_v1": [ "6.7*10^6*e^(0.0023*t)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Assuming that Switzerland's population is growing exponentially at a continuous rate of 0.15 percent a year and that the 1988 population was 6.9 million, write an expression for the population as a function of time in years. (Let $t=0$ in 1988.) $P=$ [ANS]", "answer_v2": [ "6.9*10^6*e^(0.0015*t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Assuming that Switzerland's population is growing exponentially at a continuous rate of 0.18 percent a year and that the 1988 population was 6.7 million, write an expression for the population as a function of time in years. (Let $t=0$ in 1988.) $P=$ [ANS]", "answer_v3": [ "6.7*10^6*e^(0.0018*t)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0101", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "calculus", "integral", "differential equations", "modeling", "exponentials" ], "problem_v1": "Oil is pumped continuously from a well at a rate proportional to the amount of oil left in the well. Initially there were 4 million barrels of oil in the well; six years later 2,000,000 barrels remain.\n(a) At what rate was the amount of oil in the well decreasing when there were 2,400,000 barrels remaining? rate=[ANS] barrels/year (b) When will there be 200,000 barrels remaining? after [ANS] years.", "answer_v1": [ "1/6*ln(2)*(2.4E+6)", "6*ln(20)/[ln(2)]" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Oil is pumped continuously from a well at a rate proportional to the amount of oil left in the well. Initially there were 1 million barrels of oil in the well; six years later 500,000 barrels remain.\n(a) At what rate was the amount of oil in the well decreasing when there were 600,000 barrels remaining? rate=[ANS] barrels/year (b) When will there be 50,000 barrels remaining? after [ANS] years.", "answer_v2": [ "1/6*ln(2)*600000", "6*ln(20)/[ln(2)]" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Oil is pumped continuously from a well at a rate proportional to the amount of oil left in the well. Initially there were 2 million barrels of oil in the well; six years later 1,000,000 barrels remain.\n(a) At what rate was the amount of oil in the well decreasing when there were 1,200,000 barrels remaining? rate=[ANS] barrels/year (b) When will there be 100,000 barrels remaining? after [ANS] years.", "answer_v3": [ "1/6*ln(2)*(1.2E+6)", "6*ln(20)/[ln(2)]" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0102", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "calculus", "integral", "differential equations", "modeling", "exponentials" ], "problem_v1": "The radioactive isotope carbon-14 is present in small quantities in all life forms, and it is constantly replenished until the organism dies, after which it decays to stable carbon-12 at a rate proportional to the amount of carbon-14 present, with a half-life of 5583 years. Suppose $C(t)$ is the amount of carbon-14 present at time $t$.\n(a) Find the value of the constant $k$ in the differential equation $C'=-kC$. $k=$ [ANS]\n(b) In 1988 three teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained about 91 percent of the amount of carbon-14 contained in freshly made cloth of the same material[1]. How was old the Shroud of Turin in 1988, according to these data? Age=[ANS] years [1]: The New York Times, October 18, 1988.", "answer_v1": [ "[ln(2)]/5583", "759.632" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The radioactive isotope carbon-14 is present in small quantities in all life forms, and it is constantly replenished until the organism dies, after which it decays to stable carbon-12 at a rate proportional to the amount of carbon-14 present, with a half-life of 5543 years. Suppose $C(t)$ is the amount of carbon-14 present at time $t$.\n(a) Find the value of the constant $k$ in the differential equation $C'=-kC$. $k=$ [ANS]\n(b) In 1988 three teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained about 91 percent of the amount of carbon-14 contained in freshly made cloth of the same material[1]. How was old the Shroud of Turin in 1988, according to these data? Age=[ANS] years [1]: The New York Times, October 18, 1988.", "answer_v2": [ "[ln(2)]/5543", "754.189" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The radioactive isotope carbon-14 is present in small quantities in all life forms, and it is constantly replenished until the organism dies, after which it decays to stable carbon-12 at a rate proportional to the amount of carbon-14 present, with a half-life of 5557 years. Suppose $C(t)$ is the amount of carbon-14 present at time $t$.\n(a) Find the value of the constant $k$ in the differential equation $C'=-kC$. $k=$ [ANS]\n(b) In 1988 three teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained about 91 percent of the amount of carbon-14 contained in freshly made cloth of the same material[1]. How was old the Shroud of Turin in 1988, according to these data? Age=[ANS] years [1]: The New York Times, October 18, 1988.", "answer_v3": [ "[ln(2)]/5557", "756.094" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0103", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "calculus", "integral", "differential equations", "modeling", "exponentials" ], "problem_v1": "Warfarin is a drug used as an anticoagulant. After administration of the drug is stopped, the quantity remaining in a patient's body decreases at a rate proportional to the quantity remaining. Suppose that the half-life of warfarin in the body is 38 hours. Sketch the quantity, $Q$, of warfarin in a patient's body as a function of the time, $t$ (in hours), since stopping administration of the drug. Mark 38 hours on your graph. Write a differential equation satisfied by $Q$: ${dQ\\over dt}=$ [ANS]\n(Your equation should not involve any undetermined constants.) (Your equation should not involve any undetermined constants.) How many hours does it take for the drug level in the body to be reduced to 25 percent of its original level? time=[ANS] (include)", "answer_v1": [ "-[ln(2)]*Q/38", "76" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Warfarin is a drug used as an anticoagulant. After administration of the drug is stopped, the quantity remaining in a patient's body decreases at a rate proportional to the quantity remaining. Suppose that the half-life of warfarin in the body is 35 hours. Sketch the quantity, $Q$, of warfarin in a patient's body as a function of the time, $t$ (in hours), since stopping administration of the drug. Mark 35 hours on your graph. Write a differential equation satisfied by $Q$: ${dQ\\over dt}=$ [ANS]\n(Your equation should not involve any undetermined constants.) (Your equation should not involve any undetermined constants.) How many hours does it take for the drug level in the body to be reduced to 35 percent of its original level? time=[ANS] (include)", "answer_v2": [ "-[ln(2)]*Q/35", "53.0101" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Warfarin is a drug used as an anticoagulant. After administration of the drug is stopped, the quantity remaining in a patient's body decreases at a rate proportional to the quantity remaining. Suppose that the half-life of warfarin in the body is 36 hours. Sketch the quantity, $Q$, of warfarin in a patient's body as a function of the time, $t$ (in hours), since stopping administration of the drug. Mark 36 hours on your graph. Write a differential equation satisfied by $Q$: ${dQ\\over dt}=$ [ANS]\n(Your equation should not involve any undetermined constants.) (Your equation should not involve any undetermined constants.) How many hours does it take for the drug level in the body to be reduced to 30 percent of its original level? time=[ANS] (include)", "answer_v3": [ "-[ln(2)]*Q/36", "62.5308" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0104", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "calculus", "integral", "differential equations", "modeling" ], "problem_v1": "A bank account earns 5\\% annual interest compounded continuously. Continuous payments are made out of the account at a rate of \\$11000 per year for 21 years.\n(a) Write a differential equation describing the balance $B$ in the account (where $B$ is a function of $t$, measured in years). $B'=$ [ANS]\n(b) Solve the differential equation given an initial balance of $B0$ dollars. $B=$ [ANS]\n(c) What should the initial balance be such that the account has zero balance after precisely 21 years? $B0=$ [ANS]", "answer_v1": [ "0.05*B-11000", "(B0-20000)*e^(0.05*t)+220000", "[1-e^{-0.05*21}]*220000" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "A bank account earns 2\\% annual interest compounded continuously. Continuous payments are made out of the account at a rate of \\$15000 per year for 16 years.\n(a) Write a differential equation describing the balance $B$ in the account (where $B$ is a function of $t$, measured in years). $B'=$ [ANS]\n(b) Solve the differential equation given an initial balance of $B0$ dollars. $B=$ [ANS]\n(c) What should the initial balance be such that the account has zero balance after precisely 16 years? $B0=$ [ANS]", "answer_v2": [ "0.02*B-15000", "(B0-750000)*e^(0.02*t)+\\frac{15000}{0.02}", "[1-e^{-0.02*16}]*\\frac{15000}{0.02}" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "A bank account earns 3\\% annual interest compounded continuously. Continuous payments are made out of the account at a rate of \\$11000 per year for 18 years.\n(a) Write a differential equation describing the balance $B$ in the account (where $B$ is a function of $t$, measured in years). $B'=$ [ANS]\n(b) Solve the differential equation given an initial balance of $B0$ dollars. $B=$ [ANS]\n(c) What should the initial balance be such that the account has zero balance after precisely 18 years? $B0=$ [ANS]", "answer_v3": [ "0.03*B-11000", "(B0-\\frac{11000}{0.03})*e^(0.03*t)+\\frac{11000}{0.03}", "[1-e^{-0.03*18}]*\\frac{11000}{0.03}" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0105", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "calculus", "differential equation", "exponential growth" ], "problem_v1": "The doubling period of a baterial population is $15$ minutes. At time $t=90$ minutes, the baterial population was 45760. What was the initial population at time $t=0$? [ANS]\nFind the size of the baterial population after 5 hours. [ANS]", "answer_v1": [ "715", "749731840" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The doubling period of a baterial population is $20$ minutes. At time $t=60$ minutes, the baterial population was 1696. What was the initial population at time $t=0$? [ANS]\nFind the size of the baterial population after 4 hours. [ANS]", "answer_v2": [ "212", "868352" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The doubling period of a baterial population is $15$ minutes. At time $t=60$ minutes, the baterial population was 6160. What was the initial population at time $t=0$? [ANS]\nFind the size of the baterial population after 4 hours. [ANS]", "answer_v3": [ "385", "25231360" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0106", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "calculus", "differential equation", "exponential growth", "decay" ], "problem_v1": "Some time in the future a human colony is started on Mars. The colony begins with 50000 people and grows exponentially to 250000 in 200 years. Give a formula for the size of the human population on Mars as a function of $t=$ time (in years) since the founding of the original colony [ANS]\nAssuming the population continues to grow exponentially, how long will it take to reach a size of 650000? [ANS] years What is the rate of change of the size of the population (measured in people per year) 150 years after the founding of the original colony? [ANS] people/year", "answer_v1": [ "50000*5**(t/200)", "318.738528233416", "1345.37000050225" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Some time in the future a human colony is started on Mars. The colony begins with 20000 people and grows exponentially to 140000 in 100 years. Give a formula for the size of the human population on Mars as a function of $t=$ time (in years) since the founding of the original colony [ANS]\nAssuming the population continues to grow exponentially, how long will it take to reach a size of 200000? [ANS] years What is the rate of change of the size of the population (measured in people per year) 350 years after the founding of the original colony? [ANS] people/year", "answer_v2": [ "20000*7**(t/100)", "118.329466245494", "353179.850906081" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Some time in the future a human colony is started on Mars. The colony begins with 30000 people and grows exponentially to 180000 in 150 years. Give a formula for the size of the human population on Mars as a function of $t=$ time (in years) since the founding of the original colony [ANS]\nAssuming the population continues to grow exponentially, how long will it take to reach a size of 330000? [ANS] years What is the rate of change of the size of the population (measured in people per year) 150 years after the founding of the original colony? [ANS] people/year", "answer_v3": [ "30000*6**(t/150)", "200.743624965866", "2150.11136307367" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0107", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "calculus" ], "problem_v1": "Exponential Growth and Decay: Here are a couple of examples for applications of exponentials and logarithms. You find out that in the year 1800 an ancestor of yours invested 100 dollars at 6 percent annual interest, compounded yearly. You happen to be her sole known descendant and in the year 2005 you collect the accumulated tidy sum of [ANS] dollars. You retire and devote the next 10 years of your life to writing a detailed biography of your remarkable ancestor. Strontium-90 is a biologically important radioactive isotope that is created in nuclear explosions. It has a half life of 28 years. To reduce the amount created in a particular explosion by a factor 1,000 you would have to wait [ANS] years. Round your answer to the nearest integer. Seeds found in a grave in Egypt proved to have only 84\\% of the Carbon-14 of living tissue. Those seeds were harvested [ANS] years ago. The half life of Carbon-14 is 5,730 years.", "answer_v1": [ "15406443", "279", "1441.31713488688" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Exponential Growth and Decay: Here are a couple of examples for applications of exponentials and logarithms. You find out that in the year 1800 an ancestor of yours invested 100 dollars at 6 percent annual interest, compounded yearly. You happen to be her sole known descendant and in the year 2005 you collect the accumulated tidy sum of [ANS] dollars. You retire and devote the next 10 years of your life to writing a detailed biography of your remarkable ancestor. Strontium-90 is a biologically important radioactive isotope that is created in nuclear explosions. It has a half life of 28 years. To reduce the amount created in a particular explosion by a factor 1,000 you would have to wait [ANS] years. Round your answer to the nearest integer. Seeds found in a grave in Egypt proved to have only 53\\% of the Carbon-14 of living tissue. Those seeds were harvested [ANS] years ago. The half life of Carbon-14 is 5,730 years.", "answer_v2": [ "15406443", "279", "5248.31176276204" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Exponential Growth and Decay: Here are a couple of examples for applications of exponentials and logarithms. You find out that in the year 1800 an ancestor of yours invested 100 dollars at 6 percent annual interest, compounded yearly. You happen to be her sole known descendant and in the year 2005 you collect the accumulated tidy sum of [ANS] dollars. You retire and devote the next 10 years of your life to writing a detailed biography of your remarkable ancestor. Strontium-90 is a biologically important radioactive isotope that is created in nuclear explosions. It has a half life of 28 years. To reduce the amount created in a particular explosion by a factor 1,000 you would have to wait [ANS] years. Round your answer to the nearest integer. Seeds found in a grave in Egypt proved to have only 64\\% of the Carbon-14 of living tissue. Those seeds were harvested [ANS] years ago. The half life of Carbon-14 is 5,730 years.", "answer_v3": [ "15406443", "279", "3689.29596740917" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0108", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "5", "keywords": [ "calculus" ], "problem_v1": "Human hair from a grave in Africa proved to have only 84\\% of the carbon 14 of living tissue. When was the body buried? The half life of carbon 14 is 5730 years. The body was buried about [ANS] years ago. Hint: The half-life of Carbon-14 is 5730 years. Use this and the information about 84 \\% to help you find $r$. Then find $t$.", "answer_v1": [ "1441.31713488688" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Human hair from a grave in Africa proved to have only 53\\% of the carbon 14 of living tissue. When was the body buried? The half life of carbon 14 is 5730 years. The body was buried about [ANS] years ago. Hint: The half-life of Carbon-14 is 5730 years. Use this and the information about 53 \\% to help you find $r$. Then find $t$.", "answer_v2": [ "5248.31176276204" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Human hair from a grave in Africa proved to have only 64\\% of the carbon 14 of living tissue. When was the body buried? The half life of carbon 14 is 5730 years. The body was buried about [ANS] years ago. Hint: The half-life of Carbon-14 is 5730 years. Use this and the information about 64 \\% to help you find $r$. Then find $t$.", "answer_v3": [ "3689.29596740917" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0109", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "3", "keywords": [], "problem_v1": "Suppose you want to start saving for retirement. You decide to continuously invest \\$15000 of your income each year in a risk-free investment with a 6\\% yearly interest rate, compounded continuously. If $y$ is the value of the investment, and $t$ is in years: $ \\frac{dy}{dt} =$ [ANS]\nYour answer should be in terms of $y$. You start investing at $t=0$ so $y(0)=0$. $y(t)=$ [ANS]\nWhat is the size of your investment after 30 years. $y(30)=$ [ANS]", "answer_v1": [ "0.06*y+15000", "250000*e^{0.06*t}-250000", "1.26241E+6" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose you want to start saving for retirement. You decide to continuously invest \\$25000 of your income each year in a risk-free investment with a 2\\% yearly interest rate, compounded continuously. If $y$ is the value of the investment, and $t$ is in years: $ \\frac{dy}{dt} =$ [ANS]\nYour answer should be in terms of $y$. You start investing at $t=0$ so $y(0)=0$. $y(t)=$ [ANS]\nWhat is the size of your investment after 15 years. $y(15)=$ [ANS]", "answer_v2": [ "0.02*y+25000", "(1.25E+6)*e^{0.02*t}-(1.25E+06)", "437324" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose you want to start saving for retirement. You decide to continuously invest \\$20000 of your income each year in a risk-free investment with a 3\\% yearly interest rate, compounded continuously. If $y$ is the value of the investment, and $t$ is in years: $ \\frac{dy}{dt} =$ [ANS]\nYour answer should be in terms of $y$. You start investing at $t=0$ so $y(0)=0$. $y(t)=$ [ANS]\nWhat is the size of your investment after 20 years. $y(20)=$ [ANS]", "answer_v3": [ "0.03*y+20000", "666667*e^{0.03*t}-666667", "548079" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0110", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - exponential growth & decay", "level": "2", "keywords": [], "problem_v1": "The instantaneous rate of change of the value of a certain investment (P) is proportional to its value. That is to say $ \\frac{dP}{dt} =rP$. If $r=6$ and $P(0)=4000$: $P(t)=$ [ANS].", "answer_v1": [ "4000*e^(6*t)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "MathJax.Hub.Config({MathMenu: {showContext: true}}); if(!window.MathJax) (function () {var script=document.createElement(\"script\"); script.type=\"text/javascript\"; script.src=\"https://work2.classviva.com/webwork2_files/mathjax/MathJax.js?config=TeX-MML-AM_HTMLorMML-full\"; document.getElementsByTagName(\"head\")[0].appendChild(script);})(); The instantaneous rate of change of the value of a certain investment (P) is proportional to its value. That is to say $ \\frac{dP}{dt} =rP$. If $r=9$ and $P(0)=500$: $P(t)=$ [ANS].", "answer_v2": [ "500*e^(9*t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "MathJax.Hub.Config({MathMenu: {showContext: true}}); if(!window.MathJax) (function () {var script=document.createElement(\"script\"); script.type=\"text/javascript\"; script.src=\"https://work2.classviva.com/webwork2_files/mathjax/MathJax.js?config=TeX-MML-AM_HTMLorMML-full\"; document.getElementsByTagName(\"head\")[0].appendChild(script);})(); The instantaneous rate of change of the value of a certain investment (P) is proportional to its value. That is to say $ \\frac{dP}{dt} =rP$. If $r=6$ and $P(0)=2000$: $P(t)=$ [ANS].", "answer_v3": [ "2000*e^(6*t)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0111", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - logistic", "level": "5", "keywords": [ "calculus", "differential equations", "logistic equation", "population growth", "exponential growth", "s-curve", "sigmoid" ], "problem_v1": "A rumor spreads through a school. Let $y(t)$ be the fraction of the population that has heard the rumor at time $t$ and assume that the rate at which the rumor spreads is proportional to the product of the fraction $y$ of the population that has heard the rumor and the fraction $1-y$ that has not yet heard the rumor.\nThe school has 1000 students in total. At 8 a.m., 105 students have heard the rumor, and by noon, half the school has heard it. Using the logistic model explained above, determine how much time passes before 90\\% of the students will have heard the rumor. 90\\% of the students have heard the rumor after about [ANS] hours.", "answer_v1": [ "8.10146" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A rumor spreads through a school. Let $y(t)$ be the fraction of the population that has heard the rumor at time $t$ and assume that the rate at which the rumor spreads is proportional to the product of the fraction $y$ of the population that has heard the rumor and the fraction $1-y$ that has not yet heard the rumor.\nThe school has 1000 students in total. At 8 a.m., 91 students have heard the rumor, and by noon, half the school has heard it. Using the logistic model explained above, determine how much time passes before 90\\% of the students will have heard the rumor. 90\\% of the students have heard the rumor after about [ANS] hours.", "answer_v2": [ "7.8188" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A rumor spreads through a school. Let $y(t)$ be the fraction of the population that has heard the rumor at time $t$ and assume that the rate at which the rumor spreads is proportional to the product of the fraction $y$ of the population that has heard the rumor and the fraction $1-y$ that has not yet heard the rumor.\nThe school has 1000 students in total. At 8 a.m., 96 students have heard the rumor, and by noon, half the school has heard it. Using the logistic model explained above, determine how much time passes before 90\\% of the students will have heard the rumor. 90\\% of the students have heard the rumor after about [ANS] hours.", "answer_v3": [ "7.91927" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0112", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - logistic", "level": "4", "keywords": [ "calculus", "differential equations", "logistic equation", "population growth", "exponential growth", "s-curve", "sigmoid" ], "problem_v1": "Sunset Lake is stocked with 2500 rainbow trout and after 1 year the population has grown to 6850. Assuming logistic growth with a carrying capacity of 25000, find the growth constant $k$, and determine when the population will increase to 13100. $k=$ [ANS] $\\text{yr}^{-1}$ The population will increase to 13100 after $\\approx$ [ANS] years.", "answer_v1": [ "1.2228", "1.87545" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Sunset Lake is stocked with 1100 rainbow trout and after 1 year the population has grown to 3750. Assuming logistic growth with a carrying capacity of 11000, find the growth constant $k$, and determine when the population will increase to 5600. $k=$ [ANS] $\\text{yr}^{-1}$ The population will increase to 5600 after $\\approx$ [ANS] years.", "answer_v2": [ "1.53798", "1.45229" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Sunset Lake is stocked with 1600 rainbow trout and after 1 year the population has grown to 4600. Assuming logistic growth with a carrying capacity of 16000, find the growth constant $k$, and determine when the population will increase to 8300. $k=$ [ANS] $\\text{yr}^{-1}$ The population will increase to 8300 after $\\approx$ [ANS] years.", "answer_v3": [ "1.28967", "1.76189" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0113", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - logistic", "level": "3", "keywords": [ "calculus", "differential equations", "logistic equation", "population growth", "exponential growth", "s-curve", "sigmoid" ], "problem_v1": "A population of squirrels lives in a forest with a carrying capacity of 2500. Assume logistic growth with growth constant $k=0.7 \\, \\text{yr}^{-1}$.\n(a) Find a formula for the squirrel population $P(t)$, assuming an initial population of 625 squirrels. $P(t)=$ [ANS]\n(b) How long will it take for the squirrel population to double? doubling time $\\approx$ [ANS] years", "answer_v1": [ "2500/(1-e^[(-0.7)*t]/(-0.333333))", "1.56945" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A population of squirrels lives in a forest with a carrying capacity of 1100. Assume logistic growth with growth constant $k=1 \\, \\text{yr}^{-1}$.\n(a) Find a formula for the squirrel population $P(t)$, assuming an initial population of 275 squirrels. $P(t)=$ [ANS]\n(b) How long will it take for the squirrel population to double? doubling time $\\approx$ [ANS] years", "answer_v2": [ "1100/(1-e^[(-1)*t]/(-0.333333))", "1.09861" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A population of squirrels lives in a forest with a carrying capacity of 1600. Assume logistic growth with growth constant $k=0.7 \\, \\text{yr}^{-1}$.\n(a) Find a formula for the squirrel population $P(t)$, assuming an initial population of 400 squirrels. $P(t)=$ [ANS]\n(b) How long will it take for the squirrel population to double? doubling time $\\approx$ [ANS] years", "answer_v3": [ "1600/(1-e^[(-0.7)*t]/(-0.333333))", "1.56945" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0114", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - logistic", "level": "4", "keywords": [ "differential equation' 'application' 'exponential growth' 'logistic", "Differential Equation", "logistic model" ], "problem_v1": "A population $P$ obeys the logistic model. It satisfies the equation $ \\frac{dP}{dt} = \\frac{7}{900} P (9-P)$ for $P > 0.$\n(a) The population is increasing when [ANS] $< P <$ [ANS]\n(b) The population is decreasing when $P >$ [ANS]\n(c) Assume that $P(0)=3.$ Find $P(76).$ $P(76)=$ [ANS]", "answer_v1": [ "0", "9", "9", "8.91278388687052" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A population $P$ obeys the logistic model. It satisfies the equation $ \\frac{dP}{dt} = \\frac{1}{1300} P (13-P)$ for $P > 0.$\n(a) The population is increasing when [ANS] $< P <$ [ANS]\n(b) The population is decreasing when $P >$ [ANS]\n(c) Assume that $P(0)=2.$ Find $P(57).$ $P(57)=$ [ANS]", "answer_v2": [ "0", "13", "13", "3.16271698686471" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A population $P$ obeys the logistic model. It satisfies the equation $ \\frac{dP}{dt} = \\frac{3}{1100} P (11-P)$ for $P > 0.$\n(a) The population is increasing when [ANS] $< P <$ [ANS]\n(b) The population is decreasing when $P >$ [ANS]\n(c) Assume that $P(0)=2.$ Find $P(67).$ $P(67)=$ [ANS]", "answer_v3": [ "0", "11", "11", "6.86235167362736" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0115", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - logistic", "level": "5", "keywords": [ "differential equation' 'application' 'population' 'Gompertz", "differential", "equation", "population", "Gompertz" ], "problem_v1": "Another model for a growth function for a limited population is given by the Gompertz function, which is a solution of the differential equation \\frac{dP}{dt} =c \\ln \\left( \\frac{K}{P} \\right) P where $c$ is a constant and $K$ is the carrying capacity.\n(a) Solve this differential equation for $c=0.2$, $K=3000$, and initial population $P_0=400$. $P(t)=$ [ANS].\n(b) Compute the limiting value of the size of the population. $ \\lim_{t\\to \\infty} P(t)=$ [ANS].\n(c) At what value of $P$ does $P$ grow fastest? $P=$ [ANS].", "answer_v1": [ "3000/e^(2.01490302054226*e^(- 0.2*t))", "3000", "1103.63832351433" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Another model for a growth function for a limited population is given by the Gompertz function, which is a solution of the differential equation \\frac{dP}{dt} =c \\ln \\left( \\frac{K}{P} \\right) P where $c$ is a constant and $K$ is the carrying capacity.\n(a) Solve this differential equation for $c=0.05$, $K=5000$, and initial population $P_0=100$. $P(t)=$ [ANS].\n(b) Compute the limiting value of the size of the population. $ \\lim_{t\\to \\infty} P(t)=$ [ANS].\n(c) At what value of $P$ does $P$ grow fastest? $P=$ [ANS].", "answer_v2": [ "5000/e^(3.91202300542815*e^(- 0.05*t))", "5000", "1839.39720585721" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Another model for a growth function for a limited population is given by the Gompertz function, which is a solution of the differential equation \\frac{dP}{dt} =c \\ln \\left( \\frac{K}{P} \\right) P where $c$ is a constant and $K$ is the carrying capacity.\n(a) Solve this differential equation for $c=0.1$, $K=4000$, and initial population $P_0=200$. $P(t)=$ [ANS].\n(b) Compute the limiting value of the size of the population. $ \\lim_{t\\to \\infty} P(t)=$ [ANS].\n(c) At what value of $P$ does $P$ grow fastest? $P=$ [ANS].", "answer_v3": [ "4000/e^(2.99573227355399*e^(- 0.1*t))", "4000", "1471.51776468577" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0116", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - logistic", "level": "5", "keywords": [], "problem_v1": "Let P(t) be the population of fish in a pond with carrying capacity $825$. [ANS] $=0$ $P(0)=$ [ANS]\nNote: use P,P' instead of P(t), P'(t) in your answers. After you set up your differential equation you will have to set it equal to zero so that WeBWorK will understand your answer, do this in a way so that the highest order derivative has a positive coefficient.", "answer_v1": [ "P", "80000" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let P(t) be the population of fish in a pond with carrying capacity $575$. [ANS] $=0$ $P(0)=$ [ANS]\nNote: use P,P' instead of P(t), P'(t) in your answers. After you set up your differential equation you will have to set it equal to zero so that WeBWorK will understand your answer, do this in a way so that the highest order derivative has a positive coefficient.", "answer_v2": [ "P", "15000" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let P(t) be the population of fish in a pond with carrying capacity $625$. [ANS] $=0$ $P(0)=$ [ANS]\nNote: use P,P' instead of P(t), P'(t) in your answers. After you set up your differential equation you will have to set it equal to zero so that WeBWorK will understand your answer, do this in a way so that the highest order derivative has a positive coefficient.", "answer_v3": [ "P", "35000" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0117", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - logistic", "level": "3", "keywords": [ "differential equation' 'application' 'logistic' 'model" ], "problem_v1": "$\\begin{array}{cccccccccc}\\hline t & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\ \\hline P(t) & 1007 & 1673.85 & 2470.04 & 3214.22 & 3766.59 & 4110.42 & 4301.55 & 4401.17 & 4451.35 \\\\ \\hline \\end{array}$ The table is given by logistic population growth: $ \\frac{dP}{dt} =P(a-bP)$ Find a=[ANS]\nb=[ANS]", "answer_v1": [ "0.72", "0.00016" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "$\\begin{array}{cccccccccc}\\hline t & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\ \\hline P(t) & 394 & 660.331 & 1037.66 & 1498.19 & 1968.07 & 2367.5 & 2657.66 & 2845.25 & 2957.55 \\\\ \\hline \\end{array}$ The table is given by logistic population growth: $ \\frac{dP}{dt} =P(a-bP)$ Find a=[ANS]\nb=[ANS]", "answer_v2": [ "0.62", "0.0002" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "$\\begin{array}{cccccccccc}\\hline t & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\ \\hline P(t) & 562 & 891.36 & 1329.29 & 1836.49 & 2337.96 & 2761.91 & 3075.4 & 3285 & 3415.87 \\\\ \\hline \\end{array}$ The table is given by logistic population growth: $ \\frac{dP}{dt} =P(a-bP)$ Find a=[ANS]\nb=[ANS]", "answer_v3": [ "0.576", "0.00016" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0118", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - logistic", "level": "5", "keywords": [ "calculus", "integral", "differential equations", "modeling", "population growth" ], "problem_v1": "The total number of people infected with a virus often grows like a logistic curve. Suppose that 25 people originally have the virus, and that in the early stages of the virus (with time, $t$, measured in weeks), the number of people infected is increasing exponentially with $k=1.8$. It is estimated that, in the long run, approximately 6500 people become infected.\n(a) Use this information to find a logistic function to model this situation. $P=$ [ANS]\n(b) Sketch a graph of your answer to part (a). Use your graph to estimate the length of time until the rate at which people are becoming infected starts to decrease. What is the vertical coordinate at this point? vertical coordinate=[ANS]", "answer_v1": [ "6500/[1+(6500-25)*e^(-1.8*t)/25]", "6500/2" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The total number of people infected with a virus often grows like a logistic curve. Suppose that 10 people originally have the virus, and that in the early stages of the virus (with time, $t$, measured in weeks), the number of people infected is increasing exponentially with $k=2$. It is estimated that, in the long run, approximately 4500 people become infected.\n(a) Use this information to find a logistic function to model this situation. $P=$ [ANS]\n(b) Sketch a graph of your answer to part (a). Use your graph to estimate the length of time until the rate at which people are becoming infected starts to decrease. What is the vertical coordinate at this point? vertical coordinate=[ANS]", "answer_v2": [ "4500/[1+(4500-10)*e^(-2*t)/10]", "4500/2" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The total number of people infected with a virus often grows like a logistic curve. Suppose that 15 people originally have the virus, and that in the early stages of the virus (with time, $t$, measured in weeks), the number of people infected is increasing exponentially with $k=1.8$. It is estimated that, in the long run, approximately 5000 people become infected.\n(a) Use this information to find a logistic function to model this situation. $P=$ [ANS]\n(b) Sketch a graph of your answer to part (a). Use your graph to estimate the length of time until the rate at which people are becoming infected starts to decrease. What is the vertical coordinate at this point? vertical coordinate=[ANS]", "answer_v3": [ "5000/[1+(5000-15)*e^(-1.8*t)/15]", "5000/2" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0119", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - mixing problems", "level": "5", "keywords": [ "differential equation' 'application", "Differential Equation", "Mixing problem" ], "problem_v1": "A tank contains $2520$ L of pure water. A solution that contains $0.06$ kg of sugar per liter enters a tank at the rate $6$ L/min The solution is mixed and drains from the tank at the same rate.\n(a) How much sugar is in the tank initially? [ANS] (kg)\n(b) Find the amount of sugar in the tank after $t$ minutes. amount=[ANS] (kg) (your answer should be a function of $t$)\n(c) Find the concentration of sugar in the solution in the tank after $75$ minutes. concentration=[ANS] (kg/L)", "answer_v1": [ "0", "151.2 *(1-e^{-0.00238095238095238 * t})", "(151.2 *(1- e^{-0.00238095238095238 * 75}))/2520" ], "answer_type_v1": [ "NV", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "A tank contains $1160$ L of pure water. A solution that contains $0.09$ kg of sugar per liter enters a tank at the rate $3$ L/min The solution is mixed and drains from the tank at the same rate.\n(a) How much sugar is in the tank initially? [ANS] (kg)\n(b) Find the amount of sugar in the tank after $t$ minutes. amount=[ANS] (kg) (your answer should be a function of $t$)\n(c) Find the concentration of sugar in the solution in the tank after $51$ minutes. concentration=[ANS] (kg/L)", "answer_v2": [ "0", "104.4 *(1-e^{-0.00258620689655172 * t})", "(104.4 *(1- e^{-0.00258620689655172 * 51}))/1160" ], "answer_type_v2": [ "NV", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "A tank contains $1620$ L of pure water. A solution that contains $0.06$ kg of sugar per liter enters a tank at the rate $4$ L/min The solution is mixed and drains from the tank at the same rate.\n(a) How much sugar is in the tank initially? [ANS] (kg)\n(b) Find the amount of sugar in the tank after $t$ minutes. amount=[ANS] (kg) (your answer should be a function of $t$)\n(c) Find the concentration of sugar in the solution in the tank after $63$ minutes. concentration=[ANS] (kg/L)", "answer_v3": [ "0", "97.2 *(1-e^{-0.00246913580246914 * t})", "(97.2 *(1- e^{-0.00246913580246914 * 63}))/1620" ], "answer_type_v3": [ "NV", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0120", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - mixing problems", "level": "5", "keywords": [], "problem_v1": "A tank contains $17$ gallons of water to start, $6.5$ gallons of water flow into the tank while $7$ gallons of water flow out of the tank per minute. Write a differential for the amount of water A(t) (in gallons) in the tank at time t in minutes. [ANS] $=0$ $A(0)=$ [ANS]\nSolve the differential equation: $A(t)=$ [ANS]\nNote: use A,A', etc instead of A(t), A'(t) in your answers. After you set up your differential equation you will have to set it equal to zero so that WeBWorK will understand your answer, do this in a way so that the highest order derivative has a positive coefficient.", "answer_v1": [ "A", "17", "(-0.5)*t+17" ], "answer_type_v1": [ "EX", "NV", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "A tank contains $6$ gallons of water to start, $9.5$ gallons of water flow into the tank while $3$ gallons of water flow out of the tank per minute. Write a differential for the amount of water A(t) (in gallons) in the tank at time t in minutes. [ANS] $=0$ $A(0)=$ [ANS]\nSolve the differential equation: $A(t)=$ [ANS]\nNote: use A,A', etc instead of A(t), A'(t) in your answers. After you set up your differential equation you will have to set it equal to zero so that WeBWorK will understand your answer, do this in a way so that the highest order derivative has a positive coefficient.", "answer_v2": [ "A", "6", "6.5*t+6" ], "answer_type_v2": [ "EX", "NV", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "A tank contains $10$ gallons of water to start, $7$ gallons of water flow into the tank while $4$ gallons of water flow out of the tank per minute. Write a differential for the amount of water A(t) (in gallons) in the tank at time t in minutes. [ANS] $=0$ $A(0)=$ [ANS]\nSolve the differential equation: $A(t)=$ [ANS]\nNote: use A,A', etc instead of A(t), A'(t) in your answers. After you set up your differential equation you will have to set it equal to zero so that WeBWorK will understand your answer, do this in a way so that the highest order derivative has a positive coefficient.", "answer_v3": [ "A", "10", "3*t+10" ], "answer_type_v3": [ "EX", "NV", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0121", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - mixing problems", "level": "3", "keywords": [ "differential equation' 'application' 'mixture' 'linear' 'model" ], "problem_v1": "For this problem A is the amount of salt in the tank. If a tank contains $400$ liters of liquid with $15$ grams of salt. A mixture containing $10$ grams per liter is pumped into the tank at a rate of $5$ liters/minute. The mixture is well-mixed, and pumped out at a rate of $4$ liters/minute. The amount of salt in the tank satisfies the differential equation [ANS] $=0$. Rewriting this as a linear differential equation we get [ANS] $=50$ The integrating factor is [ANS] and the solution is [ANS]", "answer_v1": [ "A", "A", "|400+1*t|^4", "50*(400+1*t)/5+[-(1.02016E+14)]*(400+1*t)^(-4)" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "For this problem A is the amount of salt in the tank. If a tank contains $125$ liters of liquid with $20$ grams of salt. A mixture containing $7$ grams per liter is pumped into the tank at a rate of $3$ liters/minute. The mixture is well-mixed, and pumped out at a rate of $6$ liters/minute. The amount of salt in the tank satisfies the differential equation [ANS] $=0$. Rewriting this as a linear differential equation we get [ANS] $=21$ The integrating factor is [ANS] and the solution is [ANS]", "answer_v2": [ "A", "A", "|125+(-3)*t|^(-2)", "21*[125+(-3)*t]/3+(-0.05472)*[125+(-3)*t]^2" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "For this problem A is the amount of salt in the tank. If a tank contains $225$ liters of liquid with $15$ grams of salt. A mixture containing $7$ grams per liter is pumped into the tank at a rate of $4$ liters/minute. The mixture is well-mixed, and pumped out at a rate of $2$ liters/minute. The amount of salt in the tank satisfies the differential equation [ANS] $=0$. Rewriting this as a linear differential equation we get [ANS] $=28$ The integrating factor is [ANS] and the solution is [ANS]", "answer_v3": [ "A", "A", "|225+2*t|^1", "28*(225+2*t)/4+(-351000)*(225+2*t)^(-1)" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0122", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - mixing problems", "level": "3", "keywords": [ "derivative" ], "problem_v1": "At time $\\small{t=0}$, a tank contains $\\small{30}$ oz of salt dissolved in $\\small{100}$ gallons of water. Then brine containing $\\small{6}$ oz of salt per gallon of brine is allowed to enter the tank at a rate of $\\small{5}$ gal/min and the mixed solution is drained from the tank at the same rate.\n(a) How much salt is in the tank at an arbitrary time? [ANS] oz. (b) How much salt is in the tank at time $\\small{20}$ min? [ANS] oz.", "answer_v1": [ "600-570*e^[-(t/20)]", "390.309" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "At time $\\small{t=0}$, a tank contains $\\small{15}$ oz of salt dissolved in $\\small{100}$ gallons of water. Then brine containing $\\small{2}$ oz of salt per gallon of brine is allowed to enter the tank at a rate of $\\small{2}$ gal/min and the mixed solution is drained from the tank at the same rate.\n(a) How much salt is in the tank at an arbitrary time? [ANS] oz. (b) How much salt is in the tank at time $\\small{30}$ min? [ANS] oz.", "answer_v2": [ "200-185*e^[-(t/50)]", "98.4698" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "At time $\\small{t=0}$, a tank contains $\\small{20}$ oz of salt dissolved in $\\small{100}$ gallons of water. Then brine containing $\\small{4}$ oz of salt per gallon of brine is allowed to enter the tank at a rate of $\\small{5}$ gal/min and the mixed solution is drained from the tank at the same rate.\n(a) How much salt is in the tank at an arbitrary time? [ANS] oz. (b) How much salt is in the tank at time $\\small{15}$ min? [ANS] oz.", "answer_v3": [ "400-380*e^[-(t/20)]", "220.501" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0123", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - mixing problems", "level": "5", "keywords": [ "differential equations", "first order", "integrating factor" ], "problem_v1": "A fish tank initially contains 30 liters of pure water. Brine of constant, but unknown, concentration of salt is flowing in at 5 liters per minute. The solution is mixed well and drained at 5 liters per minute.\nLet $x$ be the amount of salt, in grams, in the fish tank after $t$ minutes have elapsed. Find a formula for the rate of change in the amount of salt, $dx/dt$, in terms of the amount of salt in the solution $x$ and the unknown concentration of incoming brine $c$. $\\dfrac{dx}{dt}=$ [ANS] grams/minute\nFind a formula for the amount of salt, in grams, after $t$ minutes have elapsed. Your answer should be in terms of $c$ and $t$. $x(t)=$ [ANS] grams\nIn 30 minutes there are 25 grams of salt in the fish tank. What is the concentration of salt in the incoming brine? $c=$ [ANS] g/L", "answer_v1": [ "c*5-x/30*5", "c*30*[1-e^(-5*t/30)]", "25/(30*[1-e^(-5*30/30)])" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A fish tank initially contains 40 liters of pure water. Brine of constant, but unknown, concentration of salt is flowing in at 2 liters per minute. The solution is mixed well and drained at 2 liters per minute.\nLet $x$ be the amount of salt, in grams, in the fish tank after $t$ minutes have elapsed. Find a formula for the rate of change in the amount of salt, $dx/dt$, in terms of the amount of salt in the solution $x$ and the unknown concentration of incoming brine $c$. $\\dfrac{dx}{dt}=$ [ANS] grams/minute\nFind a formula for the amount of salt, in grams, after $t$ minutes have elapsed. Your answer should be in terms of $c$ and $t$. $x(t)=$ [ANS] grams\nIn 15 minutes there are 20 grams of salt in the fish tank. What is the concentration of salt in the incoming brine? $c=$ [ANS] g/L", "answer_v2": [ "c*2-x/40*2", "c*40*[1-e^(-2*t/40)]", "20/(40*[1-e^(-2*15/40)])" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A fish tank initially contains 30 liters of pure water. Brine of constant, but unknown, concentration of salt is flowing in at 3 liters per minute. The solution is mixed well and drained at 3 liters per minute.\nLet $x$ be the amount of salt, in grams, in the fish tank after $t$ minutes have elapsed. Find a formula for the rate of change in the amount of salt, $dx/dt$, in terms of the amount of salt in the solution $x$ and the unknown concentration of incoming brine $c$. $\\dfrac{dx}{dt}=$ [ANS] grams/minute\nFind a formula for the amount of salt, in grams, after $t$ minutes have elapsed. Your answer should be in terms of $c$ and $t$. $x(t)=$ [ANS] grams\nIn 15 minutes there are 25 grams of salt in the fish tank. What is the concentration of salt in the incoming brine? $c=$ [ANS] g/L", "answer_v3": [ "c*3-x/30*3", "c*30*[1-e^(-3*t/30)]", "25/(30*[1-e^(-3*15/30)])" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0124", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - mixing problems", "level": "5", "keywords": [ "differential equations", "first order", "integrating factor" ], "problem_v1": "Suppose the clean water of a stream flows into Lake Alpha, then into Lake Beta, and then further downstream. The in and out flow for each lake is 500 liters per hour. Lake Alpha contains 400 thousand liters of water, and Lake Beta contains 200 thousand liters of water. A truck with 200 kilograms of Kool-Aid drink mix crashes into Lake Alpha. Assume that the water is being continually mixed perfectly by the stream.\nLet $x$ be the amount of Kool-Aid, in kilograms, in Lake Alpha $t$ hours after the crash. Find a formula for the incremental change in the amount of Kool-Aid, $\\Delta x$, in terms of the amount of Kool-Aid in the lake $x$ and the incremental change in time $\\Delta t$. Enter $\\Delta t$ as Deltat. $\\Delta x=$ [ANS] kg\nFind a formula for the amount of Kool-Aid, in kilograms, in Lake Alpha $t$ hours after the crash. $x(t)=$ [ANS] kg\nLet $y$ be the amount of Kool-Aid, in kilograms, in Lake Beta $t$ hours after the crash. Find a formula for the incremental change in the amount of Kool-Aid, $\\Delta y$, in terms of the amounts $x, y$, and the incremental change in time $\\Delta t$. Enter $\\Delta t$ as Deltat. $\\Delta y=$ [ANS] kg\nFind a formula for the amount of Kool-Aid in Lake Beta $t$ hours after the crash. $y(t)=$ [ANS] kg", "answer_v1": [ "-(x/400000)*500*Deltat", "200*e^(-500*t/400000)", "x/400000*500*Deltat-y/200000*500*Deltat", "200*200000/(400000-200000)*[e^(-500*t/400000)-e^(-500*t/200000)]" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose the clean water of a stream flows into Lake Alpha, then into Lake Beta, and then further downstream. The in and out flow for each lake is 100 liters per hour. Lake Alpha contains 500 thousand liters of water, and Lake Beta contains 100 thousand liters of water. A truck with 200 kilograms of Kool-Aid drink mix crashes into Lake Alpha. Assume that the water is being continually mixed perfectly by the stream.\nLet $x$ be the amount of Kool-Aid, in kilograms, in Lake Alpha $t$ hours after the crash. Find a formula for the incremental change in the amount of Kool-Aid, $\\Delta x$, in terms of the amount of Kool-Aid in the lake $x$ and the incremental change in time $\\Delta t$. Enter $\\Delta t$ as Deltat. $\\Delta x=$ [ANS] kg\nFind a formula for the amount of Kool-Aid, in kilograms, in Lake Alpha $t$ hours after the crash. $x(t)=$ [ANS] kg\nLet $y$ be the amount of Kool-Aid, in kilograms, in Lake Beta $t$ hours after the crash. Find a formula for the incremental change in the amount of Kool-Aid, $\\Delta y$, in terms of the amounts $x, y$, and the incremental change in time $\\Delta t$. Enter $\\Delta t$ as Deltat. $\\Delta y=$ [ANS] kg\nFind a formula for the amount of Kool-Aid in Lake Beta $t$ hours after the crash. $y(t)=$ [ANS] kg", "answer_v2": [ "-(x/500000)*100*Deltat", "200*e^(-100*t/500000)", "x/500000*100*Deltat-y/100000*100*Deltat", "200*100000/(500000-100000)*[e^(-100*t/500000)-e^(-100*t/100000)]" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose the clean water of a stream flows into Lake Alpha, then into Lake Beta, and then further downstream. The in and out flow for each lake is 200 liters per hour. Lake Alpha contains 400 thousand liters of water, and Lake Beta contains 200 thousand liters of water. A truck with 400 kilograms of Kool-Aid drink mix crashes into Lake Alpha. Assume that the water is being continually mixed perfectly by the stream.\nLet $x$ be the amount of Kool-Aid, in kilograms, in Lake Alpha $t$ hours after the crash. Find a formula for the incremental change in the amount of Kool-Aid, $\\Delta x$, in terms of the amount of Kool-Aid in the lake $x$ and the incremental change in time $\\Delta t$. Enter $\\Delta t$ as Deltat. $\\Delta x=$ [ANS] kg\nFind a formula for the amount of Kool-Aid, in kilograms, in Lake Alpha $t$ hours after the crash. $x(t)=$ [ANS] kg\nLet $y$ be the amount of Kool-Aid, in kilograms, in Lake Beta $t$ hours after the crash. Find a formula for the incremental change in the amount of Kool-Aid, $\\Delta y$, in terms of the amounts $x, y$, and the incremental change in time $\\Delta t$. Enter $\\Delta t$ as Deltat. $\\Delta y=$ [ANS] kg\nFind a formula for the amount of Kool-Aid in Lake Beta $t$ hours after the crash. $y(t)=$ [ANS] kg", "answer_v3": [ "-(x/400000)*200*Deltat", "400*e^(-200*t/400000)", "x/400000*200*Deltat-y/200000*200*Deltat", "400*200000/(400000-200000)*[e^(-200*t/400000)-e^(-200*t/200000)]" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0125", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - circuits", "level": "4", "keywords": [ "calculus", "differential equations", "linear equation", "first order" ], "problem_v1": "Consider a series circuit consisting of a resistor of $R$ ohms, an inductor of $L$ henries, and variable voltage source of $V(t)$ volts (time $t$ in seconds). The current through the circuit $I(t)$ (in amperes) satisfies the differential equation: \\frac{dI}{dt} + \\frac{R}{L} I= \\frac{1}{L} V(t) Find the solution to this equation with the initial condition $I(0)=0$, assuming that $R=130 \\, \\Omega$, $L=5$ H, and $V(t)$ is constant with $V(t)=10$ V. $V(t)=$ [ANS]", "answer_v1": [ "0.0769231*(1-e^[(-26)*t])" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Consider a series circuit consisting of a resistor of $R$ ohms, an inductor of $L$ henries, and variable voltage source of $V(t)$ volts (time $t$ in seconds). The current through the circuit $I(t)$ (in amperes) satisfies the differential equation: \\frac{dI}{dt} + \\frac{R}{L} I= \\frac{1}{L} V(t) Find the solution to this equation with the initial condition $I(0)=0$, assuming that $R=50 \\, \\Omega$, $L=5$ H, and $V(t)$ is constant with $V(t)=10$ V. $V(t)=$ [ANS]", "answer_v2": [ "0.2*(1-e^[(-10)*t])" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Consider a series circuit consisting of a resistor of $R$ ohms, an inductor of $L$ henries, and variable voltage source of $V(t)$ volts (time $t$ in seconds). The current through the circuit $I(t)$ (in amperes) satisfies the differential equation: \\frac{dI}{dt} + \\frac{R}{L} I= \\frac{1}{L} V(t) Find the solution to this equation with the initial condition $I(0)=0$, assuming that $R=80 \\, \\Omega$, $L=5$ H, and $V(t)$ is constant with $V(t)=10$ V. $V(t)=$ [ANS]", "answer_v3": [ "0.125*(1-e^[(-16)*t])" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0126", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - circuits", "level": "5", "keywords": [ "calculus", "differential equations", "exponential growth" ], "problem_v1": "An electric current flowing in a circuit (consisting of a battery of $V$ volts and a resistor of $R$ ohms connected in series), satisfies\n \\frac{dI}{dt} =-k \\left(I-b \\right) for some constants $k$ and $b$ with $k > 0$. Initially, $I(0)=0$ and $I(t)$ approaches a maximum level $ \\frac{V}{R} $ as $t \\to \\infty$.\n(a) What is the value of $b$? $b=$ [ANS]\n(b) Find a formula for $I(t)$ in terms of $k,\\, V$, and $R$. $I(t)=$ [ANS]\n(c) What percent of its maximum value does $I(t)$ reach at time $t= \\frac{8}{k} $? [ANS] \\%", "answer_v1": [ "V/R", "V/R*[1-e^(-k*t)]", "99.9665" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "An electric current flowing in a circuit (consisting of a battery of $V$ volts and a resistor of $R$ ohms connected in series), satisfies\n \\frac{dI}{dt} =-k \\left(I-b \\right) for some constants $k$ and $b$ with $k > 0$. Initially, $I(0)=0$ and $I(t)$ approaches a maximum level $ \\frac{V}{R} $ as $t \\to \\infty$.\n(a) What is the value of $b$? $b=$ [ANS]\n(b) Find a formula for $I(t)$ in terms of $k,\\, V$, and $R$. $I(t)=$ [ANS]\n(c) What percent of its maximum value does $I(t)$ reach at time $t= \\frac{2}{k} $? [ANS] \\%", "answer_v2": [ "V/R", "V/R*[1-e^(-k*t)]", "86.4665" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "An electric current flowing in a circuit (consisting of a battery of $V$ volts and a resistor of $R$ ohms connected in series), satisfies\n \\frac{dI}{dt} =-k \\left(I-b \\right) for some constants $k$ and $b$ with $k > 0$. Initially, $I(0)=0$ and $I(t)$ approaches a maximum level $ \\frac{V}{R} $ as $t \\to \\infty$.\n(a) What is the value of $b$? $b=$ [ANS]\n(b) Find a formula for $I(t)$ in terms of $k,\\, V$, and $R$. $I(t)=$ [ANS]\n(c) What percent of its maximum value does $I(t)$ reach at time $t= \\frac{4}{k} $? [ANS] \\%", "answer_v3": [ "V/R", "V/R*[1-e^(-k*t)]", "98.1684" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0127", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - Newton's law of cooling", "level": "5", "keywords": [ "calculus", "differential equations", "exponential growth" ], "problem_v1": "A cup of coffee, cooling off in a room at temperature $23^\\circ \\text{C}$, has cooling constant $k=0.078 \\, \\text{min}^{-1}$.\n(a) How fast is the coffee cooling (in degrees per minute) when its temperature is $T=76^\\circ \\text{C}$? Cooling at the rate=[ANS] $\\frac{{}^\\circ \\text{C}} {\\text{min}}$ (b) Use the Linear Approximation to estimate the change in temperature over the next $4\\, \\text{s}$ when $T=76^\\circ \\text{C}$. Estimated to cool=[ANS] ${}^\\circ \\text{C}$ over the next $4\\, \\text{s}$ (c) The coffee is served at a temperature of $88^\\circ \\text{C}$. How long should you wait before drinking it if the optimal temperature is $63^\\circ \\text{C}$? Waiting time=[ANS] $\\text{min}$", "answer_v1": [ "4.134", "0.2756", "6.22446" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A cup of coffee, cooling off in a room at temperature $20^\\circ \\text{C}$, has cooling constant $k=0.117 \\, \\text{min}^{-1}$.\n(a) How fast is the coffee cooling (in degrees per minute) when its temperature is $T=80^\\circ \\text{C}$? Cooling at the rate=[ANS] $\\frac{{}^\\circ \\text{C}} {\\text{min}}$ (b) Use the Linear Approximation to estimate the change in temperature over the next $4\\, \\text{s}$ when $T=80^\\circ \\text{C}$. Estimated to cool=[ANS] ${}^\\circ \\text{C}$ over the next $4\\, \\text{s}$ (c) The coffee is served at a temperature of $84^\\circ \\text{C}$. How long should you wait before drinking it if the optimal temperature is $60^\\circ \\text{C}$? Waiting time=[ANS] $\\text{min}$", "answer_v2": [ "7.02", "0.468", "4.01713" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A cup of coffee, cooling off in a room at temperature $21^\\circ \\text{C}$, has cooling constant $k=0.072 \\, \\text{min}^{-1}$.\n(a) How fast is the coffee cooling (in degrees per minute) when its temperature is $T=76^\\circ \\text{C}$? Cooling at the rate=[ANS] $\\frac{{}^\\circ \\text{C}} {\\text{min}}$ (b) Use the Linear Approximation to estimate the change in temperature over the next $4\\, \\text{s}$ when $T=76^\\circ \\text{C}$. Estimated to cool=[ANS] ${}^\\circ \\text{C}$ over the next $4\\, \\text{s}$ (c) The coffee is served at a temperature of $86^\\circ \\text{C}$. How long should you wait before drinking it if the optimal temperature is $61^\\circ \\text{C}$? Waiting time=[ANS] $\\text{min}$", "answer_v3": [ "3.96", "0.264", "6.74316" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0128", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - Newton's law of cooling", "level": "5", "keywords": [ "calculus", "differential equations", "exponential growth" ], "problem_v1": "When a hot object is placed in a water bath whose temperature is $25^\\circ \\text{C}$, it cools from $100^\\circ \\text{C}$ to $50^\\circ \\text{C}$ in $190 \\text{s}$. In another bath, the same cooling occurs in $170 \\text{s}$. Find the temperature of the second bath. The temperature of the second bath=[ANS] ${}^\\circ \\text{C}$", "answer_v1": [ "20.1025" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "When a hot object is placed in a water bath whose temperature is $25^\\circ \\text{C}$, it cools from $100^\\circ \\text{C}$ to $50^\\circ \\text{C}$ in $150 \\text{s}$. In another bath, the same cooling occurs in $130 \\text{s}$. Find the temperature of the second bath. The temperature of the second bath=[ANS] ${}^\\circ \\text{C}$", "answer_v2": [ "18.5777" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "When a hot object is placed in a water bath whose temperature is $25^\\circ \\text{C}$, it cools from $100^\\circ \\text{C}$ to $50^\\circ \\text{C}$ in $165 \\text{s}$. In another bath, the same cooling occurs in $145 \\text{s}$. Find the temperature of the second bath. The temperature of the second bath=[ANS] ${}^\\circ \\text{C}$", "answer_v3": [ "19.249" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0129", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - Newton's law of cooling", "level": "5", "keywords": [ "calculus", "differential equations", "exponential growth" ], "problem_v1": "A hot metal bar is submerged in a large reservoir of water whose temperature is $70^\\circ \\text{F}$. The temperature of the bar $20\\text{s}$ after submersion is $110^\\circ \\text{F}$. After $1\\,\\text{min}$ submerged, the temperature has cooled to $90^\\circ \\text{F}$.\n(a) Determine the cooling constant $k$. $k=$ [ANS] $s^{-1}$ (b) What is the differential equation satisfied by the temperature $F(t)$ of the bar? $F'(t)=$ [ANS] Use F for F(t). Use F for F(t). (c) What is the formula for $F(t)$? $F(t)=$ [ANS]\n(d) Determine the temperature of the bar at the moment it is submerged. Initial temperature=[ANS] ${}^\\circ \\text{F}$", "answer_v1": [ "0.0173287", "(-0.0173287)*(F-70)", "70+56.5686*e^[(-0.0173287)*t]", "126.569" ], "answer_type_v1": [ "NV", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A hot metal bar is submerged in a large reservoir of water whose temperature is $40^\\circ \\text{F}$. The temperature of the bar $20\\text{s}$ after submersion is $90^\\circ \\text{F}$. After $1\\,\\text{min}$ submerged, the temperature has cooled to $70^\\circ \\text{F}$.\n(a) Determine the cooling constant $k$. $k=$ [ANS] $s^{-1}$ (b) What is the differential equation satisfied by the temperature $F(t)$ of the bar? $F'(t)=$ [ANS] Use F for F(t). Use F for F(t). (c) What is the formula for $F(t)$? $F(t)=$ [ANS]\n(d) Determine the temperature of the bar at the moment it is submerged. Initial temperature=[ANS] ${}^\\circ \\text{F}$", "answer_v2": [ "0.0127706", "(-0.0127706)*(F-40)", "40+64.5497*e^[(-0.0127706)*t]", "104.55" ], "answer_type_v2": [ "NV", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A hot metal bar is submerged in a large reservoir of water whose temperature is $50^\\circ \\text{F}$. The temperature of the bar $20\\text{s}$ after submersion is $95^\\circ \\text{F}$. After $1\\,\\text{min}$ submerged, the temperature has cooled to $75^\\circ \\text{F}$.\n(a) Determine the cooling constant $k$. $k=$ [ANS] $s^{-1}$ (b) What is the differential equation satisfied by the temperature $F(t)$ of the bar? $F'(t)=$ [ANS] Use F for F(t). Use F for F(t). (c) What is the formula for $F(t)$? $F(t)=$ [ANS]\n(d) Determine the temperature of the bar at the moment it is submerged. Initial temperature=[ANS] ${}^\\circ \\text{F}$", "answer_v3": [ "0.0146947", "(-0.0146947)*(F-50)", "50+60.3739*e^[(-0.0146947)*t]", "110.374" ], "answer_type_v3": [ "NV", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0130", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - Newton's law of cooling", "level": "4", "keywords": [ "differential equation' 'linear", "differential", "equation" ], "problem_v1": "Susan finds an alien artifact in the desert, where there are temperature variations from a low in the 30s at night to a high in the 100s in the day. She is interested in how the artifact will respond to faster variations in temperature, so she kidnaps the artifact, takes it back to her lab (hotly pursued by the military police who patrol Area 51), and sticks it in an \"oven\"--that is, a closed box whose temperature she can control precisely.\nLet $T(t)$ be the temperature of the artifact. Newton's law of cooling says that $T(t)$ changes at a rate proportional to the difference between the temperature of the environment and the temperature of the artifact. This says that there is a constant $k$, not dependent on time, such that $T'=k(E-T)$, where $E$ is the temperature of the environment (the oven).\nBefore collecting the artifact from the desert, Susan measured its temperature at a couple of times, and she has determined that for the alien artifact, $k=0.8$.\nSusan preheats her oven to $95$ degrees Fahrenheit (she has stubbornly refused to join the metric world). At time $t=0$ the oven is at exactly $95$ degrees and is heating up, and the oven runs through a temperature cycle every $2 \\pi$ minutes, in which its temperature varies by $30$ degrees above and $30$ degrees below $95$ degrees.\nLet $E(t)$ be the temperature of the oven after $t$ minutes. $E(t)=$ [ANS]\nAt time $t=0$, when the artifact is at a temperature of $70$ degrees, she puts it in the oven. Let $T(t)$ be the temperature of the artifact at time $t$. Then $T(0)=$ [ANS] (degrees)\nWrite a differential equation which models the temperature of the artifact. $T'=f(t, T)=$ R2 R2. Note: Use $T$ rather than $T(t)$ since the latter confuses the computer. Don't enter units for this equation.\nSolve the differential equation. To do this, you may find it helpful to know that if $a$ is a constant, then\n\\int \\sin(t)e^{at} dt= \\frac{1}{a^{2} +1} e^{at} (a \\sin(t)-\\cos(t))+C. $T(t)=$ [ANS]\nAfter Susan puts in the artifact in the oven, the military police break in and take her away. Think about what happens to her artifact as $t \\rightarrow \\infty$ and fill in the following sentence:\nFor large values of $t$, even though the oven temperature varies between 65 and 125 degrees, the artifact varies from [ANS] to [ANS] degrees.", "answer_v1": [ " 95 + 30*sin(t)", "70", "0.8*( 95 + 30*sin(t) - T) ", " 95 + 14.6341463414634*(0.8*sin(t)-cos(t)) - 10.3658536585366*e^(-0.8*t)", "76.2591485733673", "113.740851426633" ], "answer_type_v1": [ "EX", "NV", "EX", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Susan finds an alien artifact in the desert, where there are temperature variations from a low in the 30s at night to a high in the 100s in the day. She is interested in how the artifact will respond to faster variations in temperature, so she kidnaps the artifact, takes it back to her lab (hotly pursued by the military police who patrol Area 51), and sticks it in an \"oven\"--that is, a closed box whose temperature she can control precisely.\nLet $T(t)$ be the temperature of the artifact. Newton's law of cooling says that $T(t)$ changes at a rate proportional to the difference between the temperature of the environment and the temperature of the artifact. This says that there is a constant $k$, not dependent on time, such that $T'=k(E-T)$, where $E$ is the temperature of the environment (the oven).\nBefore collecting the artifact from the desert, Susan measured its temperature at a couple of times, and she has determined that for the alien artifact, $k=0.65$.\nSusan preheats her oven to $70$ degrees Fahrenheit (she has stubbornly refused to join the metric world). At time $t=0$ the oven is at exactly $70$ degrees and is heating up, and the oven runs through a temperature cycle every $2 \\pi$ minutes, in which its temperature varies by $35$ degrees above and $35$ degrees below $70$ degrees.\nLet $E(t)$ be the temperature of the oven after $t$ minutes. $E(t)=$ [ANS]\nAt time $t=0$, when the artifact is at a temperature of $35$ degrees, she puts it in the oven. Let $T(t)$ be the temperature of the artifact at time $t$. Then $T(0)=$ [ANS] (degrees)\nWrite a differential equation which models the temperature of the artifact. $T'=f(t, T)=$ R2 R2. Note: Use $T$ rather than $T(t)$ since the latter confuses the computer. Don't enter units for this equation.\nSolve the differential equation. To do this, you may find it helpful to know that if $a$ is a constant, then\n\\int \\sin(t)e^{at} dt= \\frac{1}{a^{2} +1} e^{at} (a \\sin(t)-\\cos(t))+C. $T(t)=$ [ANS]\nAfter Susan puts in the artifact in the oven, the military police break in and take her away. Think about what happens to her artifact as $t \\rightarrow \\infty$ and fill in the following sentence:\nFor large values of $t$, even though the oven temperature varies between 35 and 105 degrees, the artifact varies from [ANS] to [ANS] degrees.", "answer_v2": [ " 70 + 35*sin(t)", "35", "0.65*( 70 + 35*sin(t) - T) ", " 70 + 15.9929701230228*(0.65*sin(t)-cos(t)) - 19.0070298769772*e^(-0.65*t)", "50.9254077291605", "89.0745922708395" ], "answer_type_v2": [ "EX", "NV", "EX", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Susan finds an alien artifact in the desert, where there are temperature variations from a low in the 30s at night to a high in the 100s in the day. She is interested in how the artifact will respond to faster variations in temperature, so she kidnaps the artifact, takes it back to her lab (hotly pursued by the military police who patrol Area 51), and sticks it in an \"oven\"--that is, a closed box whose temperature she can control precisely.\nLet $T(t)$ be the temperature of the artifact. Newton's law of cooling says that $T(t)$ changes at a rate proportional to the difference between the temperature of the environment and the temperature of the artifact. This says that there is a constant $k$, not dependent on time, such that $T'=k(E-T)$, where $E$ is the temperature of the environment (the oven).\nBefore collecting the artifact from the desert, Susan measured its temperature at a couple of times, and she has determined that for the alien artifact, $k=0.7$.\nSusan preheats her oven to $80$ degrees Fahrenheit (she has stubbornly refused to join the metric world). At time $t=0$ the oven is at exactly $80$ degrees and is heating up, and the oven runs through a temperature cycle every $2 \\pi$ minutes, in which its temperature varies by $30$ degrees above and $30$ degrees below $80$ degrees.\nLet $E(t)$ be the temperature of the oven after $t$ minutes. $E(t)=$ [ANS]\nAt time $t=0$, when the artifact is at a temperature of $45$ degrees, she puts it in the oven. Let $T(t)$ be the temperature of the artifact at time $t$. Then $T(0)=$ [ANS] (degrees)\nWrite a differential equation which models the temperature of the artifact. $T'=f(t, T)=$ R2 R2. Note: Use $T$ rather than $T(t)$ since the latter confuses the computer. Don't enter units for this equation.\nSolve the differential equation. To do this, you may find it helpful to know that if $a$ is a constant, then\n\\int \\sin(t)e^{at} dt= \\frac{1}{a^{2} +1} e^{at} (a \\sin(t)-\\cos(t))+C. $T(t)=$ [ANS]\nAfter Susan puts in the artifact in the oven, the military police break in and take her away. Think about what happens to her artifact as $t \\rightarrow \\infty$ and fill in the following sentence:\nFor large values of $t$, even though the oven temperature varies between 50 and 110 degrees, the artifact varies from [ANS] to [ANS] degrees.", "answer_v3": [ " 80 + 30*sin(t)", "45", "0.7*( 80 + 30*sin(t) - T) ", " 80 + 14.0939597315436*(0.7*sin(t)-cos(t)) - 20.9060402684564*e^(-0.7*t)", "62.7961296691001", "97.2038703308999" ], "answer_type_v3": [ "EX", "NV", "EX", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Differential_equations_0131", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - Newton's law of cooling", "level": "5", "keywords": [ "differential equation' 'application' 'temperature" ], "problem_v1": "A thermometer is taken from a room where the temperature is $24 ^o C$ to the outdoors, where the temperature is $-3 ^o C$. After one minute the thermometer reads $14 ^o C$.\n(a) What will the reading on the thermometer be after $4$ more minutes? [ANS], (b) When will the thermometer read $-2 ^o C$? [ANS] minutes after it was taken to the outdoors.", "answer_v1": [ "-0.328288558842844", "7.12423106400972" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A thermometer is taken from a room where the temperature is $18 ^o C$ to the outdoors, where the temperature is $4 ^o C$. After one minute the thermometer reads $11 ^o C$.\n(a) What will the reading on the thermometer be after $3$ more minutes? [ANS], (b) When will the thermometer read $5 ^o C$? [ANS] minutes after it was taken to the outdoors.", "answer_v2": [ "4.875", "3.8073549220576" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A thermometer is taken from a room where the temperature is $20 ^o C$ to the outdoors, where the temperature is $-3 ^o C$. After one minute the thermometer reads $9 ^o C$.\n(a) What will the reading on the thermometer be after $4$ more minutes? [ANS], (b) When will the thermometer read $-2 ^o C$? [ANS] minutes after it was taken to the outdoors.", "answer_v3": [ "-2.11080935245371", "4.81948069577598" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0132", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - Newton's law of cooling", "level": "5", "keywords": [], "problem_v1": "A liquid at temperature $85$ F is placed in an oven at temperature $400$. The temperature of the liquid increases at a rate $6$ times the difference between the temperature of the liquid and that of the oven. Write a differential equation for the temperature T(t) of the liquid. [ANS] $=0$ $T(0)=$ [ANS]\nNote: use T,T', etc instead of T(t), T'(t) in your answers. After you set up your differential equation you will have to set it equal to zero so that WeBWorK will understand your answer, do this in a way so that the highest order derivative has a positive coefficient.", "answer_v1": [ "T", "85" ], "answer_type_v1": [ "TF", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A liquid at temperature $35$ F is placed in an oven at temperature $450$. The temperature of the liquid increases at a rate $2$ times the difference between the temperature of the liquid and that of the oven. Write a differential equation for the temperature T(t) of the liquid. [ANS] $=0$ $T(0)=$ [ANS]\nNote: use T,T', etc instead of T(t), T'(t) in your answers. After you set up your differential equation you will have to set it equal to zero so that WeBWorK will understand your answer, do this in a way so that the highest order derivative has a positive coefficient.", "answer_v2": [ "T", "35" ], "answer_type_v2": [ "TF", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A liquid at temperature $45$ F is placed in an oven at temperature $425$. The temperature of the liquid increases at a rate $3$ times the difference between the temperature of the liquid and that of the oven. Write a differential equation for the temperature T(t) of the liquid. [ANS] $=0$ $T(0)=$ [ANS]\nNote: use T,T', etc instead of T(t), T'(t) in your answers. After you set up your differential equation you will have to set it equal to zero so that WeBWorK will understand your answer, do this in a way so that the highest order derivative has a positive coefficient.", "answer_v3": [ "T", "45" ], "answer_type_v3": [ "TF", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0133", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - Newton's law of cooling", "level": "3", "keywords": [ "differential equation' 'application' 'temperature' 'linear' 'model' 'newton' 'law' 'cooling" ], "problem_v1": "For this problem T is the variable for the temperature, k is the growth constant (or constant of proportionality) and is positive. Newton's Law of cooling states that the temperature of an object changes at a rate proportional to the difference of the temperature of the surrounding medium and the temperature of the object. If the temperature of the surrounding medium is $420^\\circ F$ write a differential equation to model the situation. Differential equation: [ANS] $=0$ This is a separable differential equation: [ANS] $T^\\prime=k$ Integrating both sides with respect to t (using C as the constant of integration) we get [ANS]=[ANS]\nThe temperature of the object is given by $T=$ [ANS]", "answer_v1": [ "T", "1/(420-T)", "-[ln(|420-T|)]", "k*t+C", "420-C*e^(-k*t)" ], "answer_type_v1": [ "TF", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "For this problem T is the variable for the temperature, k is the growth constant (or constant of proportionality) and is positive. Newton's Law of cooling states that the temperature of an object changes at a rate proportional to the difference of the temperature of the surrounding medium and the temperature of the object. If the temperature of the surrounding medium is $200^\\circ F$ write a differential equation to model the situation. Differential equation: [ANS] $=0$ This is a separable differential equation: [ANS] $T^\\prime=k$ Integrating both sides with respect to t (using C as the constant of integration) we get [ANS]=[ANS]\nThe temperature of the object is given by $T=$ [ANS]", "answer_v2": [ "T", "1/(200-T)", "-[ln(|200-T|)]", "k*t+C", "200-C*e^(-k*t)" ], "answer_type_v2": [ "TF", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "For this problem T is the variable for the temperature, k is the growth constant (or constant of proportionality) and is positive. Newton's Law of cooling states that the temperature of an object changes at a rate proportional to the difference of the temperature of the surrounding medium and the temperature of the object. If the temperature of the surrounding medium is $280^\\circ F$ write a differential equation to model the situation. Differential equation: [ANS] $=0$ This is a separable differential equation: [ANS] $T^\\prime=k$ Integrating both sides with respect to t (using C as the constant of integration) we get [ANS]=[ANS]\nThe temperature of the object is given by $T=$ [ANS]", "answer_v3": [ "T", "1/(280-T)", "-[ln(|280-T|)]", "k*t+C", "280-C*e^(-k*t)" ], "answer_type_v3": [ "TF", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Differential_equations_0134", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - Newton's law of cooling", "level": "5", "keywords": [ "differential", "equation", "differential equation' 'linear" ], "problem_v1": "Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of $200$ degrees Fahrenheit when freshly poured, and $2.5$ minutes later has cooled to $183$ degrees in a room at $76$ degrees, determine when the coffee reaches a temperature of $153$ degrees. The coffee will reach a temperature of $153$ degrees in [ANS] minutes.", "answer_v1": [ "8.07845572030882" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of $210$ degrees Fahrenheit when freshly poured, and $1$ minutes later has cooled to $197$ degrees in a room at $60$ degrees, determine when the coffee reaches a temperature of $147$ degrees. The coffee will reach a temperature of $147$ degrees in [ANS] minutes.", "answer_v2": [ "6.00883538044762" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of $200$ degrees Fahrenheit when freshly poured, and $1.5$ minutes later has cooled to $184$ degrees in a room at $66$ degrees, determine when the coffee reaches a temperature of $154$ degrees. The coffee will reach a temperature of $154$ degrees in [ANS] minutes.", "answer_v3": [ "4.96050967490695" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0135", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - Newton's law of cooling", "level": "5", "keywords": [ "calculus", "integral", "differential equations", "modeling", "exponentials" ], "problem_v1": "Consider a $43^\\circ$ F object placed in $69^\\circ$ F room.\n(a) Write a differential equation for $H$, the temperature of the object at time $t$, using $k>0$ for the constant of proportionality, and write your equation in terms of $H$, $k$, and $t$. $H'=$ [ANS]\n(b) Give the general solution for your differential equation. Simplify your solution in terms of an unspecified constant $C$, which appears as the coefficient of an exponential term, and the growth factor $k$. $H=$ [ANS]\n(Your answer may involve the constant of proportionality $k$.) (c) The temperature of the object is $43^\\circ$ F initially, and $51^\\circ$ F one hour later. Find the temperature of the object after $5$ hours. $H(5)=$ [ANS] degrees F", "answer_v1": [ "k*(69-H)", "C*e^(-k*t)+69", "(43-69)*e^[5*ln((51-69)/(43-69))]+69" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Consider a $49^\\circ$ F object placed in $64^\\circ$ F room.\n(a) Write a differential equation for $H$, the temperature of the object at time $t$, using $k>0$ for the constant of proportionality, and write your equation in terms of $H$, $k$, and $t$. $H'=$ [ANS]\n(b) Give the general solution for your differential equation. Simplify your solution in terms of an unspecified constant $C$, which appears as the coefficient of an exponential term, and the growth factor $k$. $H=$ [ANS]\n(Your answer may involve the constant of proportionality $k$.) (c) The temperature of the object is $49^\\circ$ F initially, and $54^\\circ$ F one hour later. Find the temperature of the object after $4$ hours. $H(4)=$ [ANS] degrees F", "answer_v2": [ "k*(64-H)", "C*e^(-k*t)+64", "(49-64)*e^[4*ln((54-64)/(49-64))]+64" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Consider a $43^\\circ$ F object placed in $66^\\circ$ F room.\n(a) Write a differential equation for $H$, the temperature of the object at time $t$, using $k>0$ for the constant of proportionality, and write your equation in terms of $H$, $k$, and $t$. $H'=$ [ANS]\n(b) Give the general solution for your differential equation. Simplify your solution in terms of an unspecified constant $C$, which appears as the coefficient of an exponential term, and the growth factor $k$. $H=$ [ANS]\n(Your answer may involve the constant of proportionality $k$.) (c) The temperature of the object is $43^\\circ$ F initially, and $49^\\circ$ F one hour later. Find the temperature of the object after $4$ hours. $H(4)=$ [ANS] degrees F", "answer_v3": [ "k*(66-H)", "C*e^(-k*t)+66", "(43-66)*e^[4*ln((49-66)/(43-66))]+66" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0136", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - Newton's law of cooling", "level": "5", "keywords": [ "differential equation", "calculus", "exponential growth", "exponential decay" ], "problem_v1": "At 3:00 pm one winter afternoon, there is a power failure at your house in Wisconsin, and your heat does not work without electricity. When the power goes out, it is $70^{\\circ}$ F in your house. At 10:00 pm, it is $58^{\\circ}$ F in the house, and you notice that it is $8^{\\circ}$ F outside.\n(a) Assuming that the temperature, $T$, in your home obeys Newton's Law of Cooling, write the differential equation satisfied by $T$. $ \\frac{dT}{dt} =$ [ANS]\n(Use k for any constant of proportionality in your equation; your equation may involve T and the values in the problem. (b) Solve the differential equation to estimate the temperature in the house when you get up at 7:00 am the next morning. Temperature=[ANS]\nShould you worry about your water pipes freezing? [ANS] (c) Think about your equation in (a): what assumption did you make about the temperature outside? Given this (probably incorrect) assumption, would you revise your estimate up or down? (And why?) Revise [ANS]", "answer_v1": [ "k*(8-T)", "8+(70-8)*e^(-1*0.0307302*16)", "no", "down" ], "answer_type_v1": [ "EX", "EX", "TF", "MCS" ], "options_v1": [ [], [], [ "yes", "no" ], [ "up", "down" ] ], "problem_v2": "At 1:00 pm one winter afternoon, there is a power failure at your house in Wisconsin, and your heat does not work without electricity. When the power goes out, it is $72^{\\circ}$ F in your house. At 9:00 pm, it is $55^{\\circ}$ F in the house, and you notice that it is $15^{\\circ}$ F outside.\n(a) Assuming that the temperature, $T$, in your home obeys Newton's Law of Cooling, write the differential equation satisfied by $T$. $ \\frac{dT}{dt} =$ [ANS]\n(Use k for any constant of proportionality in your equation; your equation may involve T and the values in the problem. (b) Solve the differential equation to estimate the temperature in the house when you get up at 7:00 am the next morning. Temperature=[ANS]\nShould you worry about your water pipes freezing? [ANS] (c) Think about your equation in (a): what assumption did you make about the temperature outside? Given this (probably incorrect) assumption, would you revise your estimate up or down? (And why?) Revise [ANS]", "answer_v2": [ "k*(15-T)", "15+(72-15)*e^(-1*0.0442715*18)", "no", "down" ], "answer_type_v2": [ "EX", "EX", "TF", "MCS" ], "options_v2": [ [], [], [ "yes", "no" ], [ "up", "down" ] ], "problem_v3": "At 1:00 pm one winter afternoon, there is a power failure at your house in Wisconsin, and your heat does not work without electricity. When the power goes out, it is $70^{\\circ}$ F in your house. At 9:00 pm, it is $57^{\\circ}$ F in the house, and you notice that it is $7^{\\circ}$ F outside.\n(a) Assuming that the temperature, $T$, in your home obeys Newton's Law of Cooling, write the differential equation satisfied by $T$. $ \\frac{dT}{dt} =$ [ANS]\n(Use k for any constant of proportionality in your equation; your equation may involve T and the values in the problem. (b) Solve the differential equation to estimate the temperature in the house when you get up at 7:00 am the next morning. Temperature=[ANS]\nShould you worry about your water pipes freezing? [ANS] (c) Think about your equation in (a): what assumption did you make about the temperature outside? Given this (probably incorrect) assumption, would you revise your estimate up or down? (And why?) Revise [ANS]", "answer_v3": [ "k*(7-T)", "7+(70-7)*e^(-1*0.028889*18)", "no", "down" ], "answer_type_v3": [ "EX", "EX", "TF", "MCS" ], "options_v3": [ [], [], [ "yes", "no" ], [ "up", "down" ] ] }, { "id": "Differential_equations_0137", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - Newton's law of cooling", "level": "5", "keywords": [ "differential equations", "first order", "integrating factor" ], "problem_v1": "Active oxygen and free radicals are believed to be exacerbating factors in causing cell injury and aging in living tissue (see citation below). Researchers are therefore interested in understanding the protective role of natural antioxidants. In the study of one such antioxidant (Hsian-tsao leaf gum), the antioxidation activity of the substance has been found to depend on concentration in the following way:\n \\frac{dA(c)}{dc} =k[L-A(c)], \\ \\ \\ A(0)=0. In this equation, the dependent variable $A$ is a quantitative measure of antioxidant activity at concentration $c$. The constant $L$ represents a limiting or equilibrium value of this activity, and $k$ a positive rate constant.\nLet $B(c)=A(c)-L$ and reformulate the given initial value problem in terms of this new dependent variable, $B$.\n$ \\frac{dB(c)}{dc} =$ [ANS]\n$B(0)=$ [ANS]\nSolve the new initial value problem for $B(c)$ and then determine the quantity $A(c)$.\n$A(c)=$ [ANS]\nDoes the activity $A(c)$ ever exceed the value $L$? [ANS]\nDetermine the concentration at which 90\\% of the limiting antioxidation activity is achieved. (Your answer is a function of the rate constant $k$.)\n$c=$ [ANS]\nLih-Shiuh, Su-Tze Chou, and Wen-Wan Chao, \"Studies on the Antioxidative Activities of Hsian-tsao (Mesona procumbens Hemsl) Leaf Gum,\" J. Agric. Food Chem., J. Agric. Food Chem., Vol. 49, 2001, pp. 963-968.", "answer_v1": [ "-k*B", "-L", "L-L*e^(-k*c)", "no", "-[ln(0.1)]/k" ], "answer_type_v1": [ "EX", "EX", "EX", "TF", "EX" ], "options_v1": [ [], [], [], [ "yes", "no" ], [] ], "problem_v2": "Active oxygen and free radicals are believed to be exacerbating factors in causing cell injury and aging in living tissue (see citation below). Researchers are therefore interested in understanding the protective role of natural antioxidants. In the study of one such antioxidant (Hsian-tsao leaf gum), the antioxidation activity of the substance has been found to depend on concentration in the following way:\n \\frac{dA(c)}{dc} =k[L-A(c)], \\ \\ \\ A(0)=0. In this equation, the dependent variable $A$ is a quantitative measure of antioxidant activity at concentration $c$. The constant $L$ represents a limiting or equilibrium value of this activity, and $k$ a positive rate constant.\nLet $B(c)=A(c)-L$ and reformulate the given initial value problem in terms of this new dependent variable, $B$.\n$ \\frac{dB(c)}{dc} =$ [ANS]\n$B(0)=$ [ANS]\nSolve the new initial value problem for $B(c)$ and then determine the quantity $A(c)$.\n$A(c)=$ [ANS]\nDoes the activity $A(c)$ ever exceed the value $L$? [ANS]\nDetermine the concentration at which 60\\% of the limiting antioxidation activity is achieved. (Your answer is a function of the rate constant $k$.)\n$c=$ [ANS]\nLih-Shiuh, Su-Tze Chou, and Wen-Wan Chao, \"Studies on the Antioxidative Activities of Hsian-tsao (Mesona procumbens Hemsl) Leaf Gum,\" J. Agric. Food Chem., J. Agric. Food Chem., Vol. 49, 2001, pp. 963-968.", "answer_v2": [ "-k*B", "-L", "L-L*e^(-k*c)", "no", "-[ln(0.4)]/k" ], "answer_type_v2": [ "EX", "EX", "EX", "TF", "EX" ], "options_v2": [ [], [], [], [ "yes", "no" ], [] ], "problem_v3": "Active oxygen and free radicals are believed to be exacerbating factors in causing cell injury and aging in living tissue (see citation below). Researchers are therefore interested in understanding the protective role of natural antioxidants. In the study of one such antioxidant (Hsian-tsao leaf gum), the antioxidation activity of the substance has been found to depend on concentration in the following way:\n \\frac{dA(c)}{dc} =k[L-A(c)], \\ \\ \\ A(0)=0. In this equation, the dependent variable $A$ is a quantitative measure of antioxidant activity at concentration $c$. The constant $L$ represents a limiting or equilibrium value of this activity, and $k$ a positive rate constant.\nLet $B(c)=A(c)-L$ and reformulate the given initial value problem in terms of this new dependent variable, $B$.\n$ \\frac{dB(c)}{dc} =$ [ANS]\n$B(0)=$ [ANS]\nSolve the new initial value problem for $B(c)$ and then determine the quantity $A(c)$.\n$A(c)=$ [ANS]\nDoes the activity $A(c)$ ever exceed the value $L$? [ANS]\nDetermine the concentration at which 70\\% of the limiting antioxidation activity is achieved. (Your answer is a function of the rate constant $k$.)\n$c=$ [ANS]\nLih-Shiuh, Su-Tze Chou, and Wen-Wan Chao, \"Studies on the Antioxidative Activities of Hsian-tsao (Mesona procumbens Hemsl) Leaf Gum,\" J. Agric. Food Chem., J. Agric. Food Chem., Vol. 49, 2001, pp. 963-968.", "answer_v3": [ "-k*B", "-L", "L-L*e^(-k*c)", "no", "-[ln(0.3)]/k" ], "answer_type_v3": [ "EX", "EX", "EX", "TF", "EX" ], "options_v3": [ [], [], [], [ "yes", "no" ], [] ] }, { "id": "Differential_equations_0138", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [ "calculus", "differential equations", "linear equation", "first order" ], "problem_v1": "Water flows into a tank at the variable rate $R_{\\scriptsize{\\text{in}}}= \\frac{72}{1+t} $ gal/min and out at the constant rate $R_{\\scriptsize{\\text{out}}}=9$ gal/min. Let $V(t)$ be the volume of water in the tank at time $t$.\n(a) Set up a differential equation for $V(t)$ and solve it with the initial condition $V(0)=110$. $V(t)=$ [ANS]\n(b) Find the maximum value of $V$. maximum value of $V=$ [ANS] gal", "answer_v1": [ "72*ln(1+t)-9*t+110", "196.72" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Water flows into a tank at the variable rate $R_{\\scriptsize{\\text{in}}}= \\frac{50}{1+t} $ gal/min and out at the constant rate $R_{\\scriptsize{\\text{out}}}=5$ gal/min. Let $V(t)$ be the volume of water in the tank at time $t$.\n(a) Set up a differential equation for $V(t)$ and solve it with the initial condition $V(0)=80$. $V(t)=$ [ANS]\n(b) Find the maximum value of $V$. maximum value of $V=$ [ANS] gal", "answer_v2": [ "50*ln(1+t)-5*t+80", "150.129" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Water flows into a tank at the variable rate $R_{\\scriptsize{\\text{in}}}= \\frac{48}{1+t} $ gal/min and out at the constant rate $R_{\\scriptsize{\\text{out}}}=6$ gal/min. Let $V(t)$ be the volume of water in the tank at time $t$.\n(a) Set up a differential equation for $V(t)$ and solve it with the initial condition $V(0)=90$. $V(t)=$ [ANS]\n(b) Find the maximum value of $V$. maximum value of $V=$ [ANS] gal", "answer_v3": [ "48*ln(1+t)-6*t+90", "147.813" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0139", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [ "calculus", "differential equations", "exponential growth" ], "problem_v1": "A $183$-lb skydiver jumps out of an airplane (with zero initial velocity). Assume that $k=0.7$ lb-s/ft with a closed parachute and $k=5.1$ lb-s/ft with an open parachute. What is the skydiver's velocity at $t=25$ s if the parachute opens after $20$ s of free fall? velocity=[ANS] ft/s", "answer_v1": [ "-38.2312" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A $162$-lb skydiver jumps out of an airplane (with zero initial velocity). Assume that $k=0.9$ lb-s/ft with a closed parachute and $k=4.6$ lb-s/ft with an open parachute. What is the skydiver's velocity at $t=25$ s if the parachute opens after $20$ s of free fall? velocity=[ANS] ft/s", "answer_v2": [ "-36.7031" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A $169$-lb skydiver jumps out of an airplane (with zero initial velocity). Assume that $k=0.8$ lb-s/ft with a closed parachute and $k=4.8$ lb-s/ft with an open parachute. What is the skydiver's velocity at $t=25$ s if the parachute opens after $20$ s of free fall? velocity=[ANS] ft/s", "answer_v3": [ "-36.9705" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0140", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [ "calculus", "differential equations", "exponential growth" ], "problem_v1": "A $73 \\, \\text{kg}$ skydiver jumps out of an airplane. What is her terminal velocity in miles per hour (mph), assuming that $k=10\\, \\frac{\\small{\\text{kg}}}{\\small{\\text{s} }}$ for free-fall (no parachute)? terminal velocity=[ANS] mph Note: Answer should be negative for downward velocity. Note: Answer should be negative for downward velocity.", "answer_v1": [ "-160.03" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A $52 \\, \\text{kg}$ skydiver jumps out of an airplane. What is her terminal velocity in miles per hour (mph), assuming that $k=10\\, \\frac{\\small{\\text{kg}}}{\\small{\\text{s} }}$ for free-fall (no parachute)? terminal velocity=[ANS] mph Note: Answer should be negative for downward velocity. Note: Answer should be negative for downward velocity.", "answer_v2": [ "-113.994" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A $59 \\, \\text{kg}$ skydiver jumps out of an airplane. What is her terminal velocity in miles per hour (mph), assuming that $k=10\\, \\frac{\\small{\\text{kg}}}{\\small{\\text{s} }}$ for free-fall (no parachute)? terminal velocity=[ANS] mph Note: Answer should be negative for downward velocity. Note: Answer should be negative for downward velocity.", "answer_v3": [ "-129.34" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0141", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [ "differential", "equation" ], "problem_v1": "A curve passes through the point $(0,8)$ and has the property that the slope of the curve at every point $P$ is four times the $y$-coordinate of $P$. What is the equation of the curve? $y(x)=$ [ANS]", "answer_v1": [ "8 * e^(4 * x)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "A curve passes through the point $(0,2)$ and has the property that the slope of the curve at every point $P$ is five times the $y$-coordinate of $P$. What is the equation of the curve? $y(x)=$ [ANS]", "answer_v2": [ "2 * e^(5 * x)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "A curve passes through the point $(0,4)$ and has the property that the slope of the curve at every point $P$ is four times the $y$-coordinate of $P$. What is the equation of the curve? $y(x)=$ [ANS]", "answer_v3": [ "4 * e^(4 * x)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0142", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [ "differential equation' 'application" ], "problem_v1": "Here is a somewhat realistic example. You should use the phase plane plotter to look at some solutions graphically before you start solving this problem and to compare with your analytic answers to help you find errors. You will probably be surprised to find how long it takes to get all of the details of solution of a realistic problem right, even when you know how to do each of the steps.\nThere are $2520$ dollars in the bank account at the beginning of January 1990, and money is added and withdrawn from the account at a rate which follows a sinusoidal pattern, peaking in January and in July with money being added at a rate corresponding to $1880$ dollars per year, while maximum withdrawals take place at the rate of $740$ dollars per year in April and October.\nThe interest rate remains constant at the rate of $7$ percent per year, compounded continuously.\nLet $y(t)$ represents the amount of money at time $t$ ($t$ is in years). $y(0)=$ [ANS] (dollars)\nWrite a formula for the rate of deposits and withdrawals (using the functions sin(), cos() and constants): $g(t)$=[ANS]\nThe interest rate remains constant at $7$ percent per year over this period of time. With $y$ representing the amount of money in dollars at time $t$ (in years) write a differential equation which models this situation. $y'=f(t, y)=$ R2 R2. Note: Use $y$ rather than $y(t)$ since the latter confuses the computer. Don't enter units for this equation.\nFind an equation for the amount of money in the account at time $t$ where $t$ is the number of years since January 1990. $y(t)=$ [ANS]\nFind the amount of money in the bank at the beginning of January 2000 (10 years later): [ANS]\nFind a solution to the equation which does not become infinite (either positive or negative) over time: $y(t)=$ [ANS]\nDuring which months of the year does this non-growing solution have the highest values? [ANS]", "answer_v1": [ "2520", "570 + 1310 *cos(4*pi*t)", "0.07*y + 570 + 1310*cos(4*pi*t)", "10663.4*exp(0.07*t) -570/0.07 - 0.580679*cos(4*pi*t) + 104.243*sin(4*pi*t)", "13330.0889628665", " -570/0.07 - 0.580679*cos(4*pi*t) + 104.243*sin(4*pi*t)", "3" ], "answer_type_v1": [ "NV", "EX", "MCS", "EX", "NV", "EX", "NV" ], "options_v1": [ [], [], [ "1", "2", "3", "4", "5" ], [], [], [], [] ], "problem_v2": "Here is a somewhat realistic example. You should use the phase plane plotter to look at some solutions graphically before you start solving this problem and to compare with your analytic answers to help you find errors. You will probably be surprised to find how long it takes to get all of the details of solution of a realistic problem right, even when you know how to do each of the steps.\nThere are $1160$ dollars in the bank account at the beginning of January 1990, and money is added and withdrawn from the account at a rate which follows a sinusoidal pattern, peaking in January and in July with money being added at a rate corresponding to $1920$ dollars per year, while maximum withdrawals take place at the rate of $220$ dollars per year in April and October.\nThe interest rate remains constant at the rate of $4$ percent per year, compounded continuously.\nLet $y(t)$ represents the amount of money at time $t$ ($t$ is in years). $y(0)=$ [ANS] (dollars)\nWrite a formula for the rate of deposits and withdrawals (using the functions sin(), cos() and constants): $g(t)$=[ANS]\nThe interest rate remains constant at $4$ percent per year over this period of time. With $y$ representing the amount of money in dollars at time $t$ (in years) write a differential equation which models this situation. $y'=f(t, y)=$ R2 R2. Note: Use $y$ rather than $y(t)$ since the latter confuses the computer. Don't enter units for this equation.\nFind an equation for the amount of money in the account at time $t$ where $t$ is the number of years since January 1990. $y(t)=$ [ANS]\nFind the amount of money in the bank at the beginning of January 2000 (10 years later): [ANS]\nFind a solution to the equation which does not become infinite (either positive or negative) over time: $y(t)=$ [ANS]\nDuring which months of the year does this non-growing solution have the highest values? [ANS]", "answer_v2": [ "1160", "850 + 1070 *cos(4*pi*t)", "0.04*y + 850 + 1070*cos(4*pi*t)", "22410.3*exp(0.04*t) -850/0.04 - 0.271031*cos(4*pi*t) + 85.147*sin(4*pi*t)", "12181.9247740871", " -850/0.04 - 0.271031*cos(4*pi*t) + 85.147*sin(4*pi*t)", "3" ], "answer_type_v2": [ "NV", "EX", "MCS", "EX", "NV", "EX", "NV" ], "options_v2": [ [], [], [ "1", "2", "3", "4", "5" ], [], [], [], [] ], "problem_v3": "Here is a somewhat realistic example. You should use the phase plane plotter to look at some solutions graphically before you start solving this problem and to compare with your analytic answers to help you find errors. You will probably be surprised to find how long it takes to get all of the details of solution of a realistic problem right, even when you know how to do each of the steps.\nThere are $1620$ dollars in the bank account at the beginning of January 1990, and money is added and withdrawn from the account at a rate which follows a sinusoidal pattern, peaking in January and in July with money being added at a rate corresponding to $1730$ dollars per year, while maximum withdrawals take place at the rate of $550$ dollars per year in April and October.\nThe interest rate remains constant at the rate of $5$ percent per year, compounded continuously.\nLet $y(t)$ represents the amount of money at time $t$ ($t$ is in years). $y(0)=$ [ANS] (dollars)\nWrite a formula for the rate of deposits and withdrawals (using the functions sin(), cos() and constants): $g(t)$=[ANS]\nThe interest rate remains constant at $5$ percent per year over this period of time. With $y$ representing the amount of money in dollars at time $t$ (in years) write a differential equation which models this situation. $y'=f(t, y)=$ R2 R2. Note: Use $y$ rather than $y(t)$ since the latter confuses the computer. Don't enter units for this equation.\nFind an equation for the amount of money in the account at time $t$ where $t$ is the number of years since January 1990. $y(t)=$ [ANS]\nFind the amount of money in the bank at the beginning of January 2000 (10 years later): [ANS]\nFind a solution to the equation which does not become infinite (either positive or negative) over time: $y(t)=$ [ANS]\nDuring which months of the year does this non-growing solution have the highest values? [ANS]", "answer_v3": [ "1620", "590 + 1140 *cos(4*pi*t)", "0.05*y + 590 + 1140*cos(4*pi*t)", "13420.4*exp(0.05*t) -590/0.05 - 0.360951*cos(4*pi*t) + 90.7169*sin(4*pi*t)", "10326.0736093886", " -590/0.05 - 0.360951*cos(4*pi*t) + 90.7169*sin(4*pi*t)", "3" ], "answer_type_v3": [ "NV", "EX", "MCS", "EX", "NV", "EX", "NV" ], "options_v3": [ [], [], [ "1", "2", "3", "4", "5" ], [], [], [], [] ] }, { "id": "Differential_equations_0143", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [ "differential equation' 'linear" ], "problem_v1": "Use the computer to check the steps for you as you go along. There is partial credit on this problem.\nA recent college graduate borrows $90000$ dollars at an (annual) interest rate of $8.25$ per cent. Anticipating steady salary increases, the buyer expects to make payments at a monthly rate of $850 (1+t/150)$ dollars per month, where $t$ is the number of months since the loan was made.\nLet $y(t)$ be the amount of money that the graduate owes $t$ months after the loan is made. $y(0)=$ [ANS] (dollars)\nWith $y$ representing the amount of money in dollars at time $t$ (in months) write a differential equation which models this situation. $y'=f(t, y)=$ \u21d2 \u21d2. Note: Use $y$ rather than $y(t)$ since the latter confuses the computer. Remember to check your units, but don't enter units for this equation--the computer won't understand them.\nFind an equation for the amount of money owed after $t$ months. $y(t)=$ [ANS]\nNext we are going to think about how many months it will take until the loan is paid off. Remember that $y(t)$ is the amount that is owed after $t$ months. The loan is paid off when $y(t)$=[ANS]\nOnce you have calculated how many months it will take to pay off the loan, give your answer as a decimal, ignoring the fact that in real life there would be a whole number of months. To do this, you should use a graphing calculator or use to estimate the root. If you use the then once you have launched xFunctions, pull down the Multigaph Utility from the menu in the upper right hand corner, enter the function you got for $y$ (using $x$ as the independent variable, sorry!), choose appropriate ranges for the axes, and then eyeball a solution.\nThe loan will be paid off in [ANS] months.\nIf the borrower wanted the loan to be paid off in exactly $20$ years, with the same payment plan as above, how much could be borrowed? Borrowed amount=[ANS]", "answer_v1": [ "90000", "0.006875*y - 850*(1+t/150)", " 243526.170798898 + 824.242424242424*t - 153526.170798898*e**(0.006875*t)", "0", "114.882515858653", "158766.026744298" ], "answer_type_v1": [ "NV", "EX", "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Use the computer to check the steps for you as you go along. There is partial credit on this problem.\nA recent college graduate borrows $50000$ dollars at an (annual) interest rate of $9.75$ per cent. Anticipating steady salary increases, the buyer expects to make payments at a monthly rate of $650 (1+t/120)$ dollars per month, where $t$ is the number of months since the loan was made.\nLet $y(t)$ be the amount of money that the graduate owes $t$ months after the loan is made. $y(0)=$ [ANS] (dollars)\nWith $y$ representing the amount of money in dollars at time $t$ (in months) write a differential equation which models this situation. $y'=f(t, y)=$ \u21d2 \u21d2. Note: Use $y$ rather than $y(t)$ since the latter confuses the computer. Remember to check your units, but don't enter units for this equation--the computer won't understand them.\nFind an equation for the amount of money owed after $t$ months. $y(t)=$ [ANS]\nNext we are going to think about how many months it will take until the loan is paid off. Remember that $y(t)$ is the amount that is owed after $t$ months. The loan is paid off when $y(t)$=[ANS]\nOnce you have calculated how many months it will take to pay off the loan, give your answer as a decimal, ignoring the fact that in real life there would be a whole number of months. To do this, you should use a graphing calculator or use to estimate the root. If you use the then once you have launched xFunctions, pull down the Multigaph Utility from the menu in the upper right hand corner, enter the function you got for $y$ (using $x$ as the independent variable, sorry!), choose appropriate ranges for the axes, and then eyeball a solution.\nThe loan will be paid off in [ANS] months.\nIf the borrower wanted the loan to be paid off in exactly $20$ years, with the same payment plan as above, how much could be borrowed? Borrowed amount=[ANS]", "answer_v2": [ "50000", "0.008125*y - 650*(1+t/120)", " 162051.282051282 + 666.666666666667*t - 112051.282051282*e**(0.008125*t)", "0", "80.6891840319229", "116231.73395876" ], "answer_type_v2": [ "NV", "EX", "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Use the computer to check the steps for you as you go along. There is partial credit on this problem.\nA recent college graduate borrows $65000$ dollars at an (annual) interest rate of $8.5$ per cent. Anticipating steady salary increases, the buyer expects to make payments at a monthly rate of $700 (1+t/130)$ dollars per month, where $t$ is the number of months since the loan was made.\nLet $y(t)$ be the amount of money that the graduate owes $t$ months after the loan is made. $y(0)=$ [ANS] (dollars)\nWith $y$ representing the amount of money in dollars at time $t$ (in months) write a differential equation which models this situation. $y'=f(t, y)=$ \u21d2 \u21d2. Note: Use $y$ rather than $y(t)$ since the latter confuses the computer. Remember to check your units, but don't enter units for this equation--the computer won't understand them.\nFind an equation for the amount of money owed after $t$ months. $y(t)=$ [ANS]\nNext we are going to think about how many months it will take until the loan is paid off. Remember that $y(t)$ is the amount that is owed after $t$ months. The loan is paid off when $y(t)$=[ANS]\nOnce you have calculated how many months it will take to pay off the loan, give your answer as a decimal, ignoring the fact that in real life there would be a whole number of months. To do this, you should use a graphing calculator or use to estimate the root. If you use the then once you have launched xFunctions, pull down the Multigaph Utility from the menu in the upper right hand corner, enter the function you got for $y$ (using $x$ as the independent variable, sorry!), choose appropriate ranges for the axes, and then eyeball a solution.\nThe loan will be paid off in [ANS] months.\nIf the borrower wanted the loan to be paid off in exactly $20$ years, with the same payment plan as above, how much could be borrowed? Borrowed amount=[ANS]", "answer_v3": [ "65000", "0.00708333333333333*y - 700*(1+t/130)", " 206143.199361192 + 760.180995475113*t - 141143.199361192*e**(0.00708333333333333*t)", "0", "96.4285035550265", "135154.821617809" ], "answer_type_v3": [ "NV", "EX", "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Differential_equations_0144", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [ "differential equation' 'application" ], "problem_v1": "A body of mass $5$ kg is projected vertically upward with an initial velocity $77$ meters per second.\nThe gravitational constant is $g=9.8 m/s^2$. The air resistance is equal to $k|v|$ where $k$ is a constant.\nFind a formula for the velocity at any time (in terms of $k$): $v(t)=$ [ANS]\nFind the limit of this velocity for a fixed time $t_0$ as the air resistance coefficient $k$ goes to $0$. (Write $t_0$ as t0 in your answer.) $v(t_0)=$ [ANS]\nHow does this compare with the solution to the equation for velocity when there is no air resistance?\nThis illustrates an important fact, related to the fundamental theorem of ODE and called 'continuous dependence' on parameters and initial conditions. What this means is that, for a fixed time, changing the initial conditions slightly, or changing the parameters slightly, only slightly changes the value at time $t$.\nThe fact that the terminal time $t$ under consideration is a fixed, finite number is important. If you consider 'infinite' $t$, or the 'final' result you may get very different answers. Consider for example a solution to $y'=y$, whose initial condition is essentially zero, but which might vary a bit positive or negative. If the initial condition is positive the \"final\" result is plus infinity, but if the initial condition is negative the final condition is negative infinity.", "answer_v1": [ "-5*9.8/k + (77 + 5*9.8/k)*exp(-k*t/5)", "77 - t0*9.8" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "A body of mass $7$ kg is projected vertically upward with an initial velocity $12$ meters per second.\nThe gravitational constant is $g=9.8 m/s^2$. The air resistance is equal to $k|v|$ where $k$ is a constant.\nFind a formula for the velocity at any time (in terms of $k$): $v(t)=$ [ANS]\nFind the limit of this velocity for a fixed time $t_0$ as the air resistance coefficient $k$ goes to $0$. (Write $t_0$ as t0 in your answer.) $v(t_0)=$ [ANS]\nHow does this compare with the solution to the equation for velocity when there is no air resistance?\nThis illustrates an important fact, related to the fundamental theorem of ODE and called 'continuous dependence' on parameters and initial conditions. What this means is that, for a fixed time, changing the initial conditions slightly, or changing the parameters slightly, only slightly changes the value at time $t$.\nThe fact that the terminal time $t$ under consideration is a fixed, finite number is important. If you consider 'infinite' $t$, or the 'final' result you may get very different answers. Consider for example a solution to $y'=y$, whose initial condition is essentially zero, but which might vary a bit positive or negative. If the initial condition is positive the \"final\" result is plus infinity, but if the initial condition is negative the final condition is negative infinity.", "answer_v2": [ "-7*9.8/k + (12 + 7*9.8/k)*exp(-k*t/7)", "12 - t0*9.8" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "A body of mass $6$ kg is projected vertically upward with an initial velocity $34$ meters per second.\nThe gravitational constant is $g=9.8 m/s^2$. The air resistance is equal to $k|v|$ where $k$ is a constant.\nFind a formula for the velocity at any time (in terms of $k$): $v(t)=$ [ANS]\nFind the limit of this velocity for a fixed time $t_0$ as the air resistance coefficient $k$ goes to $0$. (Write $t_0$ as t0 in your answer.) $v(t_0)=$ [ANS]\nHow does this compare with the solution to the equation for velocity when there is no air resistance?\nThis illustrates an important fact, related to the fundamental theorem of ODE and called 'continuous dependence' on parameters and initial conditions. What this means is that, for a fixed time, changing the initial conditions slightly, or changing the parameters slightly, only slightly changes the value at time $t$.\nThe fact that the terminal time $t$ under consideration is a fixed, finite number is important. If you consider 'infinite' $t$, or the 'final' result you may get very different answers. Consider for example a solution to $y'=y$, whose initial condition is essentially zero, but which might vary a bit positive or negative. If the initial condition is positive the \"final\" result is plus infinity, but if the initial condition is negative the final condition is negative infinity.", "answer_v3": [ "-6*9.8/k + (34 + 6*9.8/k)*exp(-k*t/6)", "34 - t0*9.8" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0145", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [], "problem_v1": "According to Torriccelli's law the speed of water leaving the bottom of a tank is the speed that a drop of the water would have acquired if it were dropped from the height h of the water in the tank. If the acceleration due to gravity is g, the (negative) velocity of a drop of water at time t is [ANS]. Integrating we get the height of a drop of water at time t dropped from a height h [ANS]. Now find the time t when the drop of water dropped from height h hits the ground: [ANS]. Plugging this back into the equation for the velocity of the rain drop, and simplifying, we find that the speed of the drop of water dropped from height h when it hits the ground is [ANS]. By Torriccelli's law, the velocity of water leaving a tank from the bottom, where the height of the water in the tank is h, is: [ANS]\nNow we know the velocity of the water leaving the tank, if the area of the hole from which the water is leaving the tank is A, then the volume of water leaving the tank every unit of time is $V^\\prime=$ [ANS]\nNow suppose that we know the tank is rectangular with width $8$ and length $7$, then the height of the water in the tank with respect to the volume $V$ of water in the tank is h=[ANS]\nPlugging this formula for h into the formula above for $V^\\prime$, we arrive at a differential equation for the volume $V$ of the water in the tank. [ANS]=0", "answer_v1": [ "-g*t", "-g*t^2/2+h", "sqrt(2*h/g)", "sqrt(2*h*g)", "-[sqrt(2*h*g)]", "A*sqrt(2*h*g)", "V/56", "V" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "According to Torriccelli's law the speed of water leaving the bottom of a tank is the speed that a drop of the water would have acquired if it were dropped from the height h of the water in the tank. If the acceleration due to gravity is g, the (negative) velocity of a drop of water at time t is [ANS]. Integrating we get the height of a drop of water at time t dropped from a height h [ANS]. Now find the time t when the drop of water dropped from height h hits the ground: [ANS]. Plugging this back into the equation for the velocity of the rain drop, and simplifying, we find that the speed of the drop of water dropped from height h when it hits the ground is [ANS]. By Torriccelli's law, the velocity of water leaving a tank from the bottom, where the height of the water in the tank is h, is: [ANS]\nNow we know the velocity of the water leaving the tank, if the area of the hole from which the water is leaving the tank is A, then the volume of water leaving the tank every unit of time is $V^\\prime=$ [ANS]\nNow suppose that we know the tank is rectangular with width $10$ and length $3$, then the height of the water in the tank with respect to the volume $V$ of water in the tank is h=[ANS]\nPlugging this formula for h into the formula above for $V^\\prime$, we arrive at a differential equation for the volume $V$ of the water in the tank. [ANS]=0", "answer_v2": [ "-g*t", "-g*t^2/2+h", "sqrt(2*h/g)", "sqrt(2*h*g)", "-[sqrt(2*h*g)]", "A*sqrt(2*h*g)", "V/30", "V" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "According to Torriccelli's law the speed of water leaving the bottom of a tank is the speed that a drop of the water would have acquired if it were dropped from the height h of the water in the tank. If the acceleration due to gravity is g, the (negative) velocity of a drop of water at time t is [ANS]. Integrating we get the height of a drop of water at time t dropped from a height h [ANS]. Now find the time t when the drop of water dropped from height h hits the ground: [ANS]. Plugging this back into the equation for the velocity of the rain drop, and simplifying, we find that the speed of the drop of water dropped from height h when it hits the ground is [ANS]. By Torriccelli's law, the velocity of water leaving a tank from the bottom, where the height of the water in the tank is h, is: [ANS]\nNow we know the velocity of the water leaving the tank, if the area of the hole from which the water is leaving the tank is A, then the volume of water leaving the tank every unit of time is $V^\\prime=$ [ANS]\nNow suppose that we know the tank is rectangular with width $8$ and length $4$, then the height of the water in the tank with respect to the volume $V$ of water in the tank is h=[ANS]\nPlugging this formula for h into the formula above for $V^\\prime$, we arrive at a differential equation for the volume $V$ of the water in the tank. [ANS]=0", "answer_v3": [ "-g*t", "-g*t^2/2+h", "sqrt(2*h/g)", "sqrt(2*h*g)", "-[sqrt(2*h*g)]", "A*sqrt(2*h*g)", "V/32", "V" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Differential_equations_0146", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "3", "keywords": [ "differential equation' 'application' 'population' 'linear' 'model" ], "problem_v1": "The constant of proportionality is k > 0, V is the volume, S is the surface area. Remember that you must use the prime notation for derivatives, not the Leibniz notation, in all of your answers. A raindrop evaporates at a rate that is proportional to the surface area, write this as a differential equation: [ANS] $=0$.\nSince $V= \\frac{4}{3} \\pi r^3$ we have by the chain rule that $ \\frac{dV}{dt} =V^\\prime=$ [ANS]. Substituting this into the original differential equation, along with the equation $S=4\\pi r^2$, and solving for $r^\\prime$ we find that $ \\frac{dr}{dt} =$ [ANS] so that $r=$ [ANS]\nTherefore if radius is initially $6$ then $V=$ [ANS].", "answer_v1": [ "V", "4*pi*r^2*r", "-k", "-k*t+C", "1.33333*pi*(-k*t+6)^3" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "The constant of proportionality is k > 0, V is the volume, S is the surface area. Remember that you must use the prime notation for derivatives, not the Leibniz notation, in all of your answers. A raindrop evaporates at a rate that is proportional to the surface area, write this as a differential equation: [ANS] $=0$.\nSince $V= \\frac{4}{3} \\pi r^3$ we have by the chain rule that $ \\frac{dV}{dt} =V^\\prime=$ [ANS]. Substituting this into the original differential equation, along with the equation $S=4\\pi r^2$, and solving for $r^\\prime$ we find that $ \\frac{dr}{dt} =$ [ANS] so that $r=$ [ANS]\nTherefore if radius is initially $2$ then $V=$ [ANS].", "answer_v2": [ "V", "4*pi*r^2*r", "-k", "-k*t+C", "1.33333*pi*(-k*t+2)^3" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "The constant of proportionality is k > 0, V is the volume, S is the surface area. Remember that you must use the prime notation for derivatives, not the Leibniz notation, in all of your answers. A raindrop evaporates at a rate that is proportional to the surface area, write this as a differential equation: [ANS] $=0$.\nSince $V= \\frac{4}{3} \\pi r^3$ we have by the chain rule that $ \\frac{dV}{dt} =V^\\prime=$ [ANS]. Substituting this into the original differential equation, along with the equation $S=4\\pi r^2$, and solving for $r^\\prime$ we find that $ \\frac{dr}{dt} =$ [ANS] so that $r=$ [ANS]\nTherefore if radius is initially $3$ then $V=$ [ANS].", "answer_v3": [ "V", "4*pi*r^2*r", "-k", "-k*t+C", "1.33333*pi*(-k*t+3)^3" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Differential_equations_0147", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [ "calculus", "differential equations", "systems" ], "problem_v1": "Consider a conflict between two armies of $x$ and $y$ soldiers, respectively. During World War I, F. W. Lanchester assumed that if both armies are fighting a conventional battle within sight of one another, the rate at which soldiers in one army are put out of action (killed or wounded) is proportional to the amount of fire the other army can concentrate on them, which is in turn proportional to the number of soldiers in the opposing army. Thus Lanchester assumed that if there are no reinforcements and $t$ represents time since the start of the battle, then $x$ and $y$ obey the differential equations \\frac{dx}{dt} =-ay, \\qquad \\frac{dy}{dt} =-bx, where $a$ and $b$ are positive constants. Suppose that $a=0.05$ and $b=0.02$, and that the armies start with $x(0)=56$ and $y(0)=22$ thousand soldiers. (Use units of thousands of soldiers for both $x$ and $y$.)\n(a) Rewrite the system of equations as an equation for $y$ as a function of $x$: $ \\frac{dy}{dx} =$ [ANS]\n(b) Solve the differential equation you obtained in (a) to show that the equation of the phase trajectory is 0.05 y^2-0.02x^2=C, for some constant $C$. This equation is called Lanchester's square law Lanchester's square law. Given the initial conditions $x(0)=56$ and $y(0)=22$, what is $C$? $C=$ [ANS]", "answer_v1": [ "0.02*x/(0.05*y)", "0.05*22^2-0.02*56^2" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Consider a conflict between two armies of $x$ and $y$ soldiers, respectively. During World War I, F. W. Lanchester assumed that if both armies are fighting a conventional battle within sight of one another, the rate at which soldiers in one army are put out of action (killed or wounded) is proportional to the amount of fire the other army can concentrate on them, which is in turn proportional to the number of soldiers in the opposing army. Thus Lanchester assumed that if there are no reinforcements and $t$ represents time since the start of the battle, then $x$ and $y$ obey the differential equations \\frac{dx}{dt} =-ay, \\qquad \\frac{dy}{dt} =-bx, where $a$ and $b$ are positive constants. Suppose that $a=0.06$ and $b=0.01$, and that the armies start with $x(0)=43$ and $y(0)=18$ thousand soldiers. (Use units of thousands of soldiers for both $x$ and $y$.)\n(a) Rewrite the system of equations as an equation for $y$ as a function of $x$: $ \\frac{dy}{dx} =$ [ANS]\n(b) Solve the differential equation you obtained in (a) to show that the equation of the phase trajectory is 0.06 y^2-0.01x^2=C, for some constant $C$. This equation is called Lanchester's square law Lanchester's square law. Given the initial conditions $x(0)=43$ and $y(0)=18$, what is $C$? $C=$ [ANS]", "answer_v2": [ "0.01*x/(0.06*y)", "0.06*18^2-0.01*43^2" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Consider a conflict between two armies of $x$ and $y$ soldiers, respectively. During World War I, F. W. Lanchester assumed that if both armies are fighting a conventional battle within sight of one another, the rate at which soldiers in one army are put out of action (killed or wounded) is proportional to the amount of fire the other army can concentrate on them, which is in turn proportional to the number of soldiers in the opposing army. Thus Lanchester assumed that if there are no reinforcements and $t$ represents time since the start of the battle, then $x$ and $y$ obey the differential equations \\frac{dx}{dt} =-ay, \\qquad \\frac{dy}{dt} =-bx, where $a$ and $b$ are positive constants. Suppose that $a=0.05$ and $b=0.01$, and that the armies start with $x(0)=47$ and $y(0)=21$ thousand soldiers. (Use units of thousands of soldiers for both $x$ and $y$.)\n(a) Rewrite the system of equations as an equation for $y$ as a function of $x$: $ \\frac{dy}{dx} =$ [ANS]\n(b) Solve the differential equation you obtained in (a) to show that the equation of the phase trajectory is 0.05 y^2-0.01x^2=C, for some constant $C$. This equation is called Lanchester's square law Lanchester's square law. Given the initial conditions $x(0)=47$ and $y(0)=21$, what is $C$? $C=$ [ANS]", "answer_v3": [ "0.01*x/(0.05*y)", "0.05*21^2-0.01*47^2" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0148", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [ "calculus", "differential equations", "systems" ], "problem_v1": "The concentrations of two chemicals $A$ and $B$ as functions of time are denoted by $x$ and $y$ respectively. Each alone decays at a rate proportional to its concentration. Put together, they also interact to form a third substance, at a rate proportional to the product of their concentrations. All this is expressed in the equations: \\frac{dx}{dt} =-4x-x y, \\qquad \\frac{dy}{dt} =-5 y-x y. Note that $(x,y)=(0,0)$ is the only equilibrium state (both concentrations must be positive or zero).\n(a) Find an equation for $y(x)$: $ \\frac{dy}{dx} =$ [ANS]\n(b) Solve your equation to obtain a relationship between $x$ and $y$. Letting $x(0)=4$ and $y(0)=8$, give the relationship as an equation involving $x$ and $y$: [ANS]\n(Your answer will be an equation involving $x$ and $y$.) Note that one implication of your relationship should be that if $x$ and $y$ are positive and very small (so that $x$ is very much smaller than $\\ln(x)$), $y^{4}/x^{5}$ is roughly constant. (c) Suppose that $x=y$. From your equation, what are $x$ and $y$ in this case? $x=y=$ [ANS]", "answer_v1": [ "y/x*(x+5)/(y+4)", "5.38629 = y-x+ln(y^4/(x^5))", "e^[4-8-ln(8^4/(4^5))]" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "The concentrations of two chemicals $A$ and $B$ as functions of time are denoted by $x$ and $y$ respectively. Each alone decays at a rate proportional to its concentration. Put together, they also interact to form a third substance, at a rate proportional to the product of their concentrations. All this is expressed in the equations: \\frac{dx}{dt} =-2x-x y, \\qquad \\frac{dy}{dt} =-3 y-x y. Note that $(x,y)=(0,0)$ is the only equilibrium state (both concentrations must be positive or zero).\n(a) Find an equation for $y(x)$: $ \\frac{dy}{dx} =$ [ANS]\n(b) Solve your equation to obtain a relationship between $x$ and $y$. Letting $x(0)=5$ and $y(0)=10$, give the relationship as an equation involving $x$ and $y$: [ANS]\n(Your answer will be an equation involving $x$ and $y$.) Note that one implication of your relationship should be that if $x$ and $y$ are positive and very small (so that $x$ is very much smaller than $\\ln(x)$), $y^{2}/x^{3}$ is roughly constant. (c) Suppose that $x=y$. From your equation, what are $x$ and $y$ in this case? $x=y=$ [ANS]", "answer_v2": [ "y/x*(x+3)/(y+2)", "4.77686 = y-x+ln(y^2/(x^3))", "e^[5-10-ln(10^2/(5^3))]" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "The concentrations of two chemicals $A$ and $B$ as functions of time are denoted by $x$ and $y$ respectively. Each alone decays at a rate proportional to its concentration. Put together, they also interact to form a third substance, at a rate proportional to the product of their concentrations. All this is expressed in the equations: \\frac{dx}{dt} =-2x-x y, \\qquad \\frac{dy}{dt} =-3 y-x y. Note that $(x,y)=(0,0)$ is the only equilibrium state (both concentrations must be positive or zero).\n(a) Find an equation for $y(x)$: $ \\frac{dy}{dx} =$ [ANS]\n(b) Solve your equation to obtain a relationship between $x$ and $y$. Letting $x(0)=4$ and $y(0)=8$, give the relationship as an equation involving $x$ and $y$: [ANS]\n(Your answer will be an equation involving $x$ and $y$.) Note that one implication of your relationship should be that if $x$ and $y$ are positive and very small (so that $x$ is very much smaller than $\\ln(x)$), $y^{2}/x^{3}$ is roughly constant. (c) Suppose that $x=y$. From your equation, what are $x$ and $y$ in this case? $x=y=$ [ANS]", "answer_v3": [ "y/x*(x+3)/(y+2)", "4 = y-x+ln(y^2/(x^3))", "e^[4-8-ln(8^2/(4^3))]" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0149", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [ "calculus", "integral", "differential equations", "modeling" ], "problem_v1": "Water leaks from a vertical cylindrical tank through a small hole in its base at a volumetric rate proportional to the square root of the volume of water remaining. The tank initially contains 325 liters and 21 liters leak out during the first day. A. When will the tank be half empty? $t=$ [ANS]\n(include) B. How much water will remain in the tank after 4 days? volume=[ANS]\n(include)", "answer_v1": [ "8.9168505388323", "245.207850159179" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Water leaks from a vertical cylindrical tank through a small hole in its base at a volumetric rate proportional to the square root of the volume of water remaining. The tank initially contains 125 liters and 25 liters leak out during the first day. A. When will the tank be half empty? $t=$ [ANS]\n(include) B. How much water will remain in the tank after 2 days? volume=[ANS]\n(include)", "answer_v2": [ "2.77432438889846", "77.7864045000421" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Water leaks from a vertical cylindrical tank through a small hole in its base at a volumetric rate proportional to the square root of the volume of water remaining. The tank initially contains 200 liters and 21 liters leak out during the first day. A. When will the tank be half empty? $t=$ [ANS]\n(include) B. How much water will remain in the tank after 3 days? volume=[ANS]\n(include)", "answer_v3": [ "5.42841149734947", "140.49344858906" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0150", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [ "calculus", "integral", "differential equations", "modeling" ], "problem_v1": "As you know, when a course ends, students start to forget the material they have learned. One model (called the Ebbinghaus model) assumes that the rate at which a student forgets material is proportional to the difference between the material currently remembered and some positive constant, $a$. A. Let $y=f(t)$ be the fraction of the original material remembered $t$ weeks after the course has ended. Set up a differential equation for $y$, using $k$ as any constant of proportionality you may need (let $k>0$). Your equation will contain two constants; the constant $a$ (also positive) is less than $y$ for all $t$. $ \\frac{dy}{dt} =$ [ANS]\nWhat is the initial condition for your equation? $y(0)=$ [ANS]\nB. Solve the differential equation. $y=$ [ANS]\nC. What are the practical meaning (in terms of the amount remembered) of the constants in the solution $y=f(t)$? If after one week the student remembers 82 percent of the material learned in the semester, and after two weeks remembers 72 percent, how much will she or he remember after summer vacation (about 14 weeks)? percent=[ANS]", "answer_v1": [ "-k*(y-a)", "1", "(1-a)*e^(-k*t)+a", "(1-0.595)*e^(-0.587787*14)+0.595" ], "answer_type_v1": [ "EX", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "As you know, when a course ends, students start to forget the material they have learned. One model (called the Ebbinghaus model) assumes that the rate at which a student forgets material is proportional to the difference between the material currently remembered and some positive constant, $a$. A. Let $y=f(t)$ be the fraction of the original material remembered $t$ weeks after the course has ended. Set up a differential equation for $y$, using $k$ as any constant of proportionality you may need (let $k>0$). Your equation will contain two constants; the constant $a$ (also positive) is less than $y$ for all $t$. $ \\frac{dy}{dt} =$ [ANS]\nWhat is the initial condition for your equation? $y(0)=$ [ANS]\nB. Solve the differential equation. $y=$ [ANS]\nC. What are the practical meaning (in terms of the amount remembered) of the constants in the solution $y=f(t)$? If after one week the student remembers 90 percent of the material learned in the semester, and after two weeks remembers 82 percent, how much will she or he remember after summer vacation (about 14 weeks)? percent=[ANS]", "answer_v2": [ "-k*(y-a)", "1", "(1-a)*e^(-k*t)+a", "(1-0.5)*e^(-0.223144*14)+0.5" ], "answer_type_v2": [ "EX", "NV", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "As you know, when a course ends, students start to forget the material they have learned. One model (called the Ebbinghaus model) assumes that the rate at which a student forgets material is proportional to the difference between the material currently remembered and some positive constant, $a$. A. Let $y=f(t)$ be the fraction of the original material remembered $t$ weeks after the course has ended. Set up a differential equation for $y$, using $k$ as any constant of proportionality you may need (let $k>0$). Your equation will contain two constants; the constant $a$ (also positive) is less than $y$ for all $t$. $ \\frac{dy}{dt} =$ [ANS]\nWhat is the initial condition for your equation? $y(0)=$ [ANS]\nB. Solve the differential equation. $y=$ [ANS]\nC. What are the practical meaning (in terms of the amount remembered) of the constants in the solution $y=f(t)$? If after one week the student remembers 88 percent of the material learned in the semester, and after two weeks remembers 80 percent, how much will she or he remember after summer vacation (about 14 weeks)? percent=[ANS]", "answer_v3": [ "-k*(y-a)", "1", "(1-a)*e^(-k*t)+a", "(1-0.64)*e^(-0.405465*14)+0.64" ], "answer_type_v3": [ "EX", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0151", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [ "calculus", "integral", "differential equations", "modeling" ], "problem_v1": "Dead leaves accumulate on the ground in a forest at a rate of 5 grams per square centimeter per year. At the same time, these leaves decompose at a continuous rate of 70 percent per year. A. Write a differential equation for the total quantity $Q$ of dead leaves (per square centimeter) at time $t$: ${dQ\\over dt}=$ [ANS]\nB. Sketch a solution to your differential equation showing that the quantity of dead leaves tends toward an equilibrium level. Assume that initially ($t=0$) there are no leaves on the ground. What is the initial quantity of leaves? $Q(0)=$ [ANS]\nWhat is the equilibrium level? $Q_{eq}=$ [ANS]\nDoes the equilibrium value attained depend on the initial condition? [ANS] A. yes B. no", "answer_v1": [ "5 - 0.7*Q", "0", "5/0.7", "B" ], "answer_type_v1": [ "EX", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B" ] ], "problem_v2": "Dead leaves accumulate on the ground in a forest at a rate of 2 grams per square centimeter per year. At the same time, these leaves decompose at a continuous rate of 85 percent per year. A. Write a differential equation for the total quantity $Q$ of dead leaves (per square centimeter) at time $t$: ${dQ\\over dt}=$ [ANS]\nB. Sketch a solution to your differential equation showing that the quantity of dead leaves tends toward an equilibrium level. Assume that initially ($t=0$) there are no leaves on the ground. What is the initial quantity of leaves? $Q(0)=$ [ANS]\nWhat is the equilibrium level? $Q_{eq}=$ [ANS]\nDoes the equilibrium value attained depend on the initial condition? [ANS] A. yes B. no", "answer_v2": [ "2 - 0.85*Q", "0", "2/0.85", "B" ], "answer_type_v2": [ "EX", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B" ] ], "problem_v3": "Dead leaves accumulate on the ground in a forest at a rate of 3 grams per square centimeter per year. At the same time, these leaves decompose at a continuous rate of 70 percent per year. A. Write a differential equation for the total quantity $Q$ of dead leaves (per square centimeter) at time $t$: ${dQ\\over dt}=$ [ANS]\nB. Sketch a solution to your differential equation showing that the quantity of dead leaves tends toward an equilibrium level. Assume that initially ($t=0$) there are no leaves on the ground. What is the initial quantity of leaves? $Q(0)=$ [ANS]\nWhat is the equilibrium level? $Q_{eq}=$ [ANS]\nDoes the equilibrium value attained depend on the initial condition? [ANS] A. yes B. no", "answer_v3": [ "3 - 0.7*Q", "0", "3/0.7", "B" ], "answer_type_v3": [ "EX", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B" ] ] }, { "id": "Differential_equations_0152", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [ "calculus", "integral", "differential equations", "modeling", "exponentials" ], "problem_v1": "Hydrocodone bitartrate is used as a cough suppressant. After the drug is fully absorbed, the quantity of drug in the body decreases at a rate proportional to the amount left in the body, with constant of proportionality $k < 0$. Suppose that the half-life of hydrocodone bitartrate in the body is 3.9 hours, and that the oral dose taken is 10 mg.\n(a) Write a differential equation for the quantity, $Q$, of hydrocodone bitartrate in the body at time $t$, in hours, since the drug was fully absorbed. $Q'=$ [ANS]\n(Write your answer in terms of $Q$ and $t$, e.g., $Q'=3 t(1-k Q)$.) (b) Solve your differential equation, assuming that at $t=0$ the patient has just absorbed the full 10 mg dose of the drug. $Q=$ [ANS]\n(Your solution may include the constant of proporationality $k$, but should not include any other unspecified constant.) (c) Use the half-life to find the constant of proportionality, $k$. $k=$ [ANS]\n(d) How much of the 10 mg dose is still in the body after 12 hours? Amount=[ANS]\n(include)", "answer_v1": [ "k*Q", "10*e^(k*t)", "[ln(1/2)]/3.9", "1.1851" ], "answer_type_v1": [ "EX", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Hydrocodone bitartrate is used as a cough suppressant. After the drug is fully absorbed, the quantity of drug in the body decreases at a rate proportional to the amount left in the body, with constant of proportionality $k < 0$. Suppose that the half-life of hydrocodone bitartrate in the body is 3.6 hours, and that the oral dose taken is 12 mg.\n(a) Write a differential equation for the quantity, $Q$, of hydrocodone bitartrate in the body at time $t$, in hours, since the drug was fully absorbed. $Q'=$ [ANS]\n(Write your answer in terms of $Q$ and $t$, e.g., $Q'=3 t(1-k Q)$.) (b) Solve your differential equation, assuming that at $t=0$ the patient has just absorbed the full 12 mg dose of the drug. $Q=$ [ANS]\n(Your solution may include the constant of proporationality $k$, but should not include any other unspecified constant.) (c) Use the half-life to find the constant of proportionality, $k$. $k=$ [ANS]\n(d) How much of the 12 mg dose is still in the body after 12 hours? Amount=[ANS]\n(include)", "answer_v2": [ "k*Q", "12*e^(k*t)", "[ln(1/2)]/3.6", "1.19055" ], "answer_type_v2": [ "EX", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Hydrocodone bitartrate is used as a cough suppressant. After the drug is fully absorbed, the quantity of drug in the body decreases at a rate proportional to the amount left in the body, with constant of proportionality $k < 0$. Suppose that the half-life of hydrocodone bitartrate in the body is 3.7 hours, and that the oral dose taken is 11 mg.\n(a) Write a differential equation for the quantity, $Q$, of hydrocodone bitartrate in the body at time $t$, in hours, since the drug was fully absorbed. $Q'=$ [ANS]\n(Write your answer in terms of $Q$ and $t$, e.g., $Q'=3 t(1-k Q)$.) (b) Solve your differential equation, assuming that at $t=0$ the patient has just absorbed the full 11 mg dose of the drug. $Q=$ [ANS]\n(Your solution may include the constant of proporationality $k$, but should not include any other unspecified constant.) (c) Use the half-life to find the constant of proportionality, $k$. $k=$ [ANS]\n(d) How much of the 11 mg dose is still in the body after 12 hours? Amount=[ANS]\n(include)", "answer_v3": [ "k*Q", "11*e^(k*t)", "[ln(1/2)]/3.7", "1.16166" ], "answer_type_v3": [ "EX", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0153", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [ "calculus", "integral", "differential equations", "modeling" ], "problem_v1": "Water leaks out of a barrel so that the rate of change in the water level is proportional to the square root of the depth of the water at that time. If the water level starts at 34 inches and drops to 33 inches in 1 hour, how many hours will it take for all of the water to leak out of the barrel? number of hours=[ANS]", "answer_v1": [ "[sqrt(34)]/[sqrt(34)-sqrt(33)]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Water leaks out of a barrel so that the rate of change in the water level is proportional to the square root of the depth of the water at that time. If the water level starts at 25 inches and drops to 24 inches in 1 hour, how many hours will it take for all of the water to leak out of the barrel? number of hours=[ANS]", "answer_v2": [ "[sqrt(25)]/[sqrt(25)-sqrt(24)]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Water leaks out of a barrel so that the rate of change in the water level is proportional to the square root of the depth of the water at that time. If the water level starts at 28 inches and drops to 27 inches in 1 hour, how many hours will it take for all of the water to leak out of the barrel? number of hours=[ANS]", "answer_v3": [ "[sqrt(28)]/[sqrt(28)-sqrt(27)]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0154", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [ "differential equations", "applications", "calculus", "modeling" ], "problem_v1": "According to a simple physiological model, an athletic adult male needs 20 calories per day per pound of body weight to maintain his weight. If he consumes more or fewer calories than those required to maintain his weight, his weight changes at a rate proportional to the difference between the number of calories consumed and the number needed to maintain his current weight; the constant of proportionality is $1/3500$ pounds per calorie. Suppose that a particular person has a constant caloric intake of $H$ calories per day. Let $W(t)$ be the person's weight in pounds at time $t$ (measured in days).\n(a) What differential equation has solution $W(t)$? $ \\frac{dW}{dt} =$ [ANS]\n(Your answer may involve W, H and values given in the problem.) (b) Solve this differential equation, if the person starts out weighing 175 pounds and consumes 3600 calories a day. $W=$ [ANS]\n(c) What happens to the person's weight as $t\\to\\infty$? $W \\to$ [ANS]", "answer_v1": [ "1/3500*(H-20*W)", "3600/20+(175-3600/20)*e^(-t/175)", "3600/20" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "According to a simple physiological model, an athletic adult male needs 20 calories per day per pound of body weight to maintain his weight. If he consumes more or fewer calories than those required to maintain his weight, his weight changes at a rate proportional to the difference between the number of calories consumed and the number needed to maintain his current weight; the constant of proportionality is $1/3500$ pounds per calorie. Suppose that a particular person has a constant caloric intake of $H$ calories per day. Let $W(t)$ be the person's weight in pounds at time $t$ (measured in days).\n(a) What differential equation has solution $W(t)$? $ \\frac{dW}{dt} =$ [ANS]\n(Your answer may involve W, H and values given in the problem.) (b) Solve this differential equation, if the person starts out weighing 150 pounds and consumes 3500 calories a day. $W=$ [ANS]\n(c) What happens to the person's weight as $t\\to\\infty$? $W \\to$ [ANS]", "answer_v2": [ "1/3500*(H-20*W)", "3500/20+(150-3500/20)*e^(-t/175)", "3500/20" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "According to a simple physiological model, an athletic adult male needs 20 calories per day per pound of body weight to maintain his weight. If he consumes more or fewer calories than those required to maintain his weight, his weight changes at a rate proportional to the difference between the number of calories consumed and the number needed to maintain his current weight; the constant of proportionality is $1/3500$ pounds per calorie. Suppose that a particular person has a constant caloric intake of $H$ calories per day. Let $W(t)$ be the person's weight in pounds at time $t$ (measured in days).\n(a) What differential equation has solution $W(t)$? $ \\frac{dW}{dt} =$ [ANS]\n(Your answer may involve W, H and values given in the problem.) (b) Solve this differential equation, if the person starts out weighing 160 pounds and consumes 3300 calories a day. $W=$ [ANS]\n(c) What happens to the person's weight as $t\\to\\infty$? $W \\to$ [ANS]", "answer_v3": [ "1/3500*(H-20*W)", "3300/20+(160-3300/20)*e^(-t/175)", "3300/20" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0155", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "3", "keywords": [ "Differantial Equations", "Separation of Variables", "General Solution", "Specific Solution" ], "problem_v1": "A rocket, fired from rest at time $\\small{t=0}$, has an initial mass of $\\small{m0}$ (including its fuel). Assuming that the fuel is consumed at a constant rate $\\small{k}$, the mass $\\small{m}$ of the rocket, while fuel is being burned, will be given by $\\small{m0-kt}$. It can be shown that if air resistance is neglected and the fuel gases are expelled at a constant speed $\\small{c}$ relative to the rocket, then the velocity $\\small{v}$ of the rocket will satisfy the equation \\small{m \\frac{dv}{dt} =ck-mg} where $\\small{g}$ is the acceleration due to gravity.\n(a) Find $\\small{v(t)}$ keeping in mind that the mass $\\small{m}$ is a function of $\\small{t}$. $\\small{v(t)}$=[ANS] m/sec (b) Suppose that the fuel accounts for 75\\% of the initial mass of the rocket and that all of the fuel is consumed at 110 s. Find the velocity of the rocket in meters per second at the instant the fuel is exhausted. [Note: Note: Take $\\small{g=9.8 \\;m/s^2}$ and $\\small{c=2500 \\;m/s}$.] $\\small{v(110)}$=[ANS] m/sec [Round to nearest whole number]", "answer_v1": [ "c*ln(m0/(m0-k*t))-g*t", "2500*ln(1/0.25)-9.8*110" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A rocket, fired from rest at time $\\small{t=0}$, has an initial mass of $\\small{m0}$ (including its fuel). Assuming that the fuel is consumed at a constant rate $\\small{k}$, the mass $\\small{m}$ of the rocket, while fuel is being burned, will be given by $\\small{m0-kt}$. It can be shown that if air resistance is neglected and the fuel gases are expelled at a constant speed $\\small{c}$ relative to the rocket, then the velocity $\\small{v}$ of the rocket will satisfy the equation \\small{m \\frac{dv}{dt} =ck-mg} where $\\small{g}$ is the acceleration due to gravity.\n(a) Find $\\small{v(t)}$ keeping in mind that the mass $\\small{m}$ is a function of $\\small{t}$. $\\small{v(t)}$=[ANS] m/sec (b) Suppose that the fuel accounts for 50\\% of the initial mass of the rocket and that all of the fuel is consumed at 150 s. Find the velocity of the rocket in meters per second at the instant the fuel is exhausted. [Note: Note: Take $\\small{g=9.8 \\;m/s^2}$ and $\\small{c=2500 \\;m/s}$.] $\\small{v(150)}$=[ANS] m/sec [Round to nearest whole number]", "answer_v2": [ "c*ln(m0/(m0-k*t))-g*t", "2500*ln(1/0.5)-9.8*150" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A rocket, fired from rest at time $\\small{t=0}$, has an initial mass of $\\small{m0}$ (including its fuel). Assuming that the fuel is consumed at a constant rate $\\small{k}$, the mass $\\small{m}$ of the rocket, while fuel is being burned, will be given by $\\small{m0-kt}$. It can be shown that if air resistance is neglected and the fuel gases are expelled at a constant speed $\\small{c}$ relative to the rocket, then the velocity $\\small{v}$ of the rocket will satisfy the equation \\small{m \\frac{dv}{dt} =ck-mg} where $\\small{g}$ is the acceleration due to gravity.\n(a) Find $\\small{v(t)}$ keeping in mind that the mass $\\small{m}$ is a function of $\\small{t}$. $\\small{v(t)}$=[ANS] m/sec (b) Suppose that the fuel accounts for 60\\% of the initial mass of the rocket and that all of the fuel is consumed at 110 s. Find the velocity of the rocket in meters per second at the instant the fuel is exhausted. [Note: Note: Take $\\small{g=9.8 \\;m/s^2}$ and $\\small{c=2500 \\;m/s}$.] $\\small{v(110)}$=[ANS] m/sec [Round to nearest whole number]", "answer_v3": [ "c*ln(m0/(m0-k*t))-g*t", "2500*ln(1/0.4)-9.8*110" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0156", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "3", "keywords": [], "problem_v1": "Suppose the population $P$ of rainbow trout in a fish hatchery is modeled by the differential equation \\frac{dP}{dt} =P (6-P), where $P$ is measured in thousands of trout and $t$ is measured in years. Suppose $P(0)=1$.\n(a) How many trout are initially in the hatchery? [ANS] trout.\n(b) Find a formula for the population at any time. $P(t)=$ [ANS] thousands of trout.\n(c) What is the size of the trout population after a very long time? $ \\lim_{t \\to \\infty} P(t)=$ [ANS] thousands of trout.\n(d) At what time is the trout population increasing most rapidly? $t=$ [ANS] years.", "answer_v1": [ "1000", "6*e^(6*t)/[e^(6*t)+5]", "6", "[ln(5)]/6" ], "answer_type_v1": [ "NV", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose the population $P$ of rainbow trout in a fish hatchery is modeled by the differential equation \\frac{dP}{dt} =P (3-P), where $P$ is measured in thousands of trout and $t$ is measured in years. Suppose $P(0)=1$.\n(a) How many trout are initially in the hatchery? [ANS] trout.\n(b) Find a formula for the population at any time. $P(t)=$ [ANS] thousands of trout.\n(c) What is the size of the trout population after a very long time? $ \\lim_{t \\to \\infty} P(t)=$ [ANS] thousands of trout.\n(d) At what time is the trout population increasing most rapidly? $t=$ [ANS] years.", "answer_v2": [ "1000", "3*e^(3*t)/[e^(3*t)+2]", "3", "[ln(2)]/3" ], "answer_type_v2": [ "NV", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose the population $P$ of rainbow trout in a fish hatchery is modeled by the differential equation \\frac{dP}{dt} =P (4-P), where $P$ is measured in thousands of trout and $t$ is measured in years. Suppose $P(0)=1$.\n(a) How many trout are initially in the hatchery? [ANS] trout.\n(b) Find a formula for the population at any time. $P(t)=$ [ANS] thousands of trout.\n(c) What is the size of the trout population after a very long time? $ \\lim_{t \\to \\infty} P(t)=$ [ANS] thousands of trout.\n(d) At what time is the trout population increasing most rapidly? $t=$ [ANS] years.", "answer_v3": [ "1000", "4*e^(4*t)/[e^(4*t)+3]", "4", "[ln(3)]/4" ], "answer_type_v3": [ "NV", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0157", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [ "calculus" ], "problem_v1": "Suppose that news spreads through a city of fixed size of 800000 people at a time rate proportional to the number of people who have not heard the news. (a.) Formulate a differential equation and initial condition for $y(t)$, the number of people who have heard the news $t$ days after it has happened. No one has heard the news at first, so $y(0)=0$. The \"time rate of increase in the number of people who have heard the news is proportional to the number of people who have not heard the news\" translates into the differential equation $ \\frac{dy}{dt} =k ($ [ANS] $)$, where $k$ is the proportionality constant. (b.) 7 days after a scandal in City Hall was reported, a poll showed that 400000 people have heard the news. Using this information and the differential equation, solve for the number of people who have heard the news after $t$ days. $y(t)=$ [ANS]", "answer_v1": [ "800000-y", "800000*[1-exp(ln(0.5)*t/7)]" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Suppose that news spreads through a city of fixed size of 200000 people at a time rate proportional to the number of people who have not heard the news. (a.) Formulate a differential equation and initial condition for $y(t)$, the number of people who have heard the news $t$ days after it has happened. No one has heard the news at first, so $y(0)=0$. The \"time rate of increase in the number of people who have heard the news is proportional to the number of people who have not heard the news\" translates into the differential equation $ \\frac{dy}{dt} =k ($ [ANS] $)$, where $k$ is the proportionality constant. (b.) 9 days after a scandal in City Hall was reported, a poll showed that 100000 people have heard the news. Using this information and the differential equation, solve for the number of people who have heard the news after $t$ days. $y(t)=$ [ANS]", "answer_v2": [ "200000-y", "200000*[1-exp(ln(0.5)*t/9)]" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Suppose that news spreads through a city of fixed size of 400000 people at a time rate proportional to the number of people who have not heard the news. (a.) Formulate a differential equation and initial condition for $y(t)$, the number of people who have heard the news $t$ days after it has happened. No one has heard the news at first, so $y(0)=0$. The \"time rate of increase in the number of people who have heard the news is proportional to the number of people who have not heard the news\" translates into the differential equation $ \\frac{dy}{dt} =k ($ [ANS] $)$, where $k$ is the proportionality constant. (b.) 8 days after a scandal in City Hall was reported, a poll showed that 200000 people have heard the news. Using this information and the differential equation, solve for the number of people who have heard the news after $t$ days. $y(t)=$ [ANS]", "answer_v3": [ "400000-y", "400000*[1-exp(ln(0.5)*t/8)]" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0158", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Applications - other", "level": "5", "keywords": [ "differential equations", "first order", "separable differential equations" ], "problem_v1": "Let $Q(t)$ represent the amount of a certain reactant present at time $t$. Suppose that the rate of decrease of $Q(t)$ is proportional to $Q^3(t)$. That is, $Q^{\\,\\prime}=-kQ^3$, where $k$ is a positive constant of proportionality.\nSuppose $ Q(0)= \\frac{1}{8} $. How long will it take for the reactant to be reduced to one half of its original amount?\n$t=$ [ANS]\nSuppose $ Q(0)= \\frac{1}{4} $. How long will it take for the reactant to be reduced to one half of its original amount?\n$t=$ [ANS]\nRecall that, in problems of radioactive decay where the differential equation has the form $Q^{\\,\\prime}=-kQ$, the half-life was independent of the amount of radioactive material initially present. What happens in the case of $Q^{\\,\\prime}=-kQ^3$? Does half-life depend on $Q(0)$, the amount initially present? [ANS]", "answer_v1": [ "96/k", "24/k", "yes" ], "answer_type_v1": [ "EX", "EX", "MCS" ], "options_v1": [ [], [], [ "yes", "no", "cannot be determined" ] ], "problem_v2": "Let $Q(t)$ represent the amount of a certain reactant present at time $t$. Suppose that the rate of decrease of $Q(t)$ is proportional to $Q^3(t)$. That is, $Q^{\\,\\prime}=-kQ^3$, where $k$ is a positive constant of proportionality.\nSuppose $ Q(0)= \\frac{1}{2} $. How long will it take for the reactant to be reduced to one half of its original amount?\n$t=$ [ANS]\nSuppose $ Q(0)= \\frac{1}{8} $. How long will it take for the reactant to be reduced to one half of its original amount?\n$t=$ [ANS]\nRecall that, in problems of radioactive decay where the differential equation has the form $Q^{\\,\\prime}=-kQ$, the half-life was independent of the amount of radioactive material initially present. What happens in the case of $Q^{\\,\\prime}=-kQ^3$? Does half-life depend on $Q(0)$, the amount initially present? [ANS]", "answer_v2": [ "6/k", "96/k", "yes" ], "answer_type_v2": [ "EX", "EX", "MCS" ], "options_v2": [ [], [], [ "yes", "no", "cannot be determined" ] ], "problem_v3": "Let $Q(t)$ represent the amount of a certain reactant present at time $t$. Suppose that the rate of decrease of $Q(t)$ is proportional to $Q^3(t)$. That is, $Q^{\\,\\prime}=-kQ^3$, where $k$ is a positive constant of proportionality.\nSuppose $ Q(0)= \\frac{1}{2} $. How long will it take for the reactant to be reduced to one half of its original amount?\n$t=$ [ANS]\nSuppose $ Q(0)= \\frac{1}{4} $. How long will it take for the reactant to be reduced to one half of its original amount?\n$t=$ [ANS]\nRecall that, in problems of radioactive decay where the differential equation has the form $Q^{\\,\\prime}=-kQ$, the half-life was independent of the amount of radioactive material initially present. What happens in the case of $Q^{\\,\\prime}=-kQ^3$? Does half-life depend on $Q(0)$, the amount initially present? [ANS]", "answer_v3": [ "6/k", "24/k", "yes" ], "answer_type_v3": [ "EX", "EX", "MCS" ], "options_v3": [ [], [], [ "yes", "no", "cannot be determined" ] ] }, { "id": "Differential_equations_0159", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Direction fields", "level": "5", "keywords": [ "tangent' 'slope' 'family of curves", "Orthogonal trajectories" ], "problem_v1": "Consider the curves in the first quadrant that have equations y=A \\exp(6x), where $A$ is a positive constant. Different values of $A$ give different curves. The curves form a family, $F$. Let $P=(6,6).$ Let $C$ be the member of the family $F$ that goes through P. A. Let $y=f(x)$ be the equation of $C$. Find $f(x)$. $f(x)=$ [ANS]\nB. Find the slope at $P$ of the tangent to $C$. slope=[ANS]\nC. A curve $D$ is perpendicular to $C$ at $P$. What is the slope of the tangent to $D$ at the point $P$? slope=[ANS]\nD. Give a formula $g(y)$ for the slope at $(x,y)$ of the member of $F$ that goes through $(x,y)$. The formula should not involve $A$ or $x$. $g(y)=$ [ANS]\nE. A curve which at each of its points is perpendicular to the member of the family $F$ that goes through that point is called an orthogonal trajectory to $F$. Each orthogonal trajectory to $F$ satisfies the differential equation\n \\frac{dy}{dx} =- \\frac{1}{g(y)} where $g(y)$ is the answer to part D. Find a function $h(y)$ such that $x=h(y)$ is the equation of the orthogonal trajectory to $F$ that passes through the point $P$. $h(y)=$ [ANS]", "answer_v1": [ "6 * e^(6 * (x - 6))", "36", "-0.0277777777777778", "6 * y", "6 + (6 /2)*((6 *6 ) - y*y)" ], "answer_type_v1": [ "EX", "NV", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Consider the curves in the first quadrant that have equations y=A \\exp(2x), where $A$ is a positive constant. Different values of $A$ give different curves. The curves form a family, $F$. Let $P=(9,2).$ Let $C$ be the member of the family $F$ that goes through P. A. Let $y=f(x)$ be the equation of $C$. Find $f(x)$. $f(x)=$ [ANS]\nB. Find the slope at $P$ of the tangent to $C$. slope=[ANS]\nC. A curve $D$ is perpendicular to $C$ at $P$. What is the slope of the tangent to $D$ at the point $P$? slope=[ANS]\nD. Give a formula $g(y)$ for the slope at $(x,y)$ of the member of $F$ that goes through $(x,y)$. The formula should not involve $A$ or $x$. $g(y)=$ [ANS]\nE. A curve which at each of its points is perpendicular to the member of the family $F$ that goes through that point is called an orthogonal trajectory to $F$. Each orthogonal trajectory to $F$ satisfies the differential equation\n \\frac{dy}{dx} =- \\frac{1}{g(y)} where $g(y)$ is the answer to part D. Find a function $h(y)$ such that $x=h(y)$ is the equation of the orthogonal trajectory to $F$ that passes through the point $P$. $h(y)=$ [ANS]", "answer_v2": [ "2 * e^(2 * (x - 9))", "4", "-0.25", "2 * y", "9 + (2 /2)*((2 *2 ) - y*y)" ], "answer_type_v2": [ "EX", "NV", "NV", "EX", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Consider the curves in the first quadrant that have equations y=A \\exp(3x), where $A$ is a positive constant. Different values of $A$ give different curves. The curves form a family, $F$. Let $P=(6,3).$ Let $C$ be the member of the family $F$ that goes through P. A. Let $y=f(x)$ be the equation of $C$. Find $f(x)$. $f(x)=$ [ANS]\nB. Find the slope at $P$ of the tangent to $C$. slope=[ANS]\nC. A curve $D$ is perpendicular to $C$ at $P$. What is the slope of the tangent to $D$ at the point $P$? slope=[ANS]\nD. Give a formula $g(y)$ for the slope at $(x,y)$ of the member of $F$ that goes through $(x,y)$. The formula should not involve $A$ or $x$. $g(y)=$ [ANS]\nE. A curve which at each of its points is perpendicular to the member of the family $F$ that goes through that point is called an orthogonal trajectory to $F$. Each orthogonal trajectory to $F$ satisfies the differential equation\n \\frac{dy}{dx} =- \\frac{1}{g(y)} where $g(y)$ is the answer to part D. Find a function $h(y)$ such that $x=h(y)$ is the equation of the orthogonal trajectory to $F$ that passes through the point $P$. $h(y)=$ [ANS]", "answer_v3": [ "3 * e^(3 * (x - 6))", "9", "-0.111111111111111", "3 * y", "6 + (3 /2)*((3 *3 ) - y*y)" ], "answer_type_v3": [ "EX", "NV", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Differential_equations_0160", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Integrating factor", "level": "4", "keywords": [ "calculus", "differential equations", "linear equation", "first order" ], "problem_v1": "Solve the initial value problem $y^{'}+(\\tan x)y=8 \\cos x, \\qquad y(0)=2$ $y(x)=$ [ANS]", "answer_v1": [ "(8*x+2)*cos(x)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the initial value problem $y^{'}+(\\tan x)y=2 \\cos x, \\qquad y(0)=8$ $y(x)=$ [ANS]", "answer_v2": [ "(2*x+8)*cos(x)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the initial value problem $y^{'}+(\\tan x)y=4 \\cos x, \\qquad y(0)=2$ $y(x)=$ [ANS]", "answer_v3": [ "(4*x+2)*cos(x)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0161", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Integrating factor", "level": "3", "keywords": [ "calculus", "differential equations", "linear equation", "first order" ], "problem_v1": "Solve $x y'=8 y-6x, \\qquad y(1)=2$.\n(a) Identify the integrating factor, $\\alpha(x)$. $\\alpha(x)$=[ANS]\n(b) Find the general solution. $y(x)=$ [ANS]\nNote: Use $C$ for the arbitrary constant. (c) Solve the initial value problem $y(1)=2$. $y(x)=$ [ANS]", "answer_v1": [ "x^(-8)", "6*x/7+C*x^8", "6*x/7+1.14286*x^8" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Solve $x y'=2 y-9x, \\qquad y(1)=-7$.\n(a) Identify the integrating factor, $\\alpha(x)$. $\\alpha(x)$=[ANS]\n(b) Find the general solution. $y(x)=$ [ANS]\nNote: Use $C$ for the arbitrary constant. (c) Solve the initial value problem $y(1)=-7$. $y(x)=$ [ANS]", "answer_v2": [ "x^(-2)", "9*x+C*x^2", "9*x-16*x^2" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Solve $x y'=4 y-6x, \\qquad y(1)=-4$.\n(a) Identify the integrating factor, $\\alpha(x)$. $\\alpha(x)$=[ANS]\n(b) Find the general solution. $y(x)=$ [ANS]\nNote: Use $C$ for the arbitrary constant. (c) Solve the initial value problem $y(1)=-4$. $y(x)=$ [ANS]", "answer_v3": [ "x^(-4)", "6*x/3+C*x^4", "6*x/3-6*x^4" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0162", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Integrating factor", "level": "4", "keywords": [ "calculus", "differential equations", "linear equation", "first order" ], "problem_v1": "Find the general solution of the first-order linear differential equation y'-(\\ln x)y=8x^x. $y(x)=$ [ANS]\nNote: Use $C$ for the arbitrary constant.", "answer_v1": [ "8*x^x+C*x^x*e^(-x)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the general solution of the first-order linear differential equation y'-(\\ln x)y=2x^x. $y(x)=$ [ANS]\nNote: Use $C$ for the arbitrary constant.", "answer_v2": [ "2*x^x+C*x^x*e^(-x)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the general solution of the first-order linear differential equation y'-(\\ln x)y=4x^x. $y(x)=$ [ANS]\nNote: Use $C$ for the arbitrary constant.", "answer_v3": [ "4*x^x+C*x^x*e^(-x)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0163", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Integrating factor", "level": "3", "keywords": [ "differential", "equation" ], "problem_v1": "Find the function $y(t)$ that satisfies the differential equation \\frac{dy}{dt} -2 t y=9 t^2 e^{t^2} and the condition $y(0)=1$. $y(t)=$ [ANS].", "answer_v1": [ "(3*t^3 + 1)*2.71828182845905^(t^2)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the function $y(t)$ that satisfies the differential equation \\frac{dy}{dt} -2 t y=-15 t^2 e^{t^2} and the condition $y(0)=5$. $y(t)=$ [ANS].", "answer_v2": [ "(-5*t^3 + 5)*2.71828182845905^(t^2)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the function $y(t)$ that satisfies the differential equation \\frac{dy}{dt} -2 t y=-6 t^2 e^{t^2} and the condition $y(0)=1$. $y(t)=$ [ANS].", "answer_v3": [ "(-2*t^3 + 1)*2.71828182845905^(t^2)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0164", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Integrating factor", "level": "3", "keywords": [ "differential", "equation" ], "problem_v1": "Solve the initial value problem \\frac{dx}{dt} +5x=\\cos(4 t) with $x(0)=2$. $x(t)=$ [ANS].", "answer_v1": [ "1.8780487804878*2.71828182845905^(- 5*t) + 5/41*cos(4*t) + 4/41*sin(4*t)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the initial value problem \\frac{dx}{dt} +2x=\\cos(2 t) with $x(0)=-2$. $x(t)=$ [ANS].", "answer_v2": [ "-2.25*2.71828182845905^(- 2*t) + 2/8*cos(2*t) + 2/8*sin(2*t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the initial value problem \\frac{dx}{dt} +3x=\\cos(3 t) with $x(0)=1$. $x(t)=$ [ANS].", "answer_v3": [ "0.833333333333333*2.71828182845905^(- 3*t) + 3/18*cos(3*t) + 3/18*sin(3*t)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0165", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Integrating factor", "level": "3", "keywords": [ "differential", "equation", "initial", "solution" ], "problem_v1": "Find the function satisfying the differential equation f'(t)-f(t)=8 t and the condition $f(2)=4$. $f(t)=$ [ANS].", "answer_v1": [ "3.78938793062516*2.71828182845905**t - 8*t - 8" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the function satisfying the differential equation f'(t)-f(t)=2 t and the condition $f(1)=-3$. $f(t)=$ [ANS].", "answer_v2": [ "0.367879441171442*2.71828182845905**t - 2*t - 2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the function satisfying the differential equation f'(t)-f(t)=4 t and the condition $f(1)=1$. $f(t)=$ [ANS].", "answer_v3": [ "3.31091497054298*2.71828182845905**t - 4*t - 4" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0166", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Integrating factor", "level": "3", "keywords": [ "differential equations", "separable", "differential equation' 'linear" ], "problem_v1": "Solve the initial value problem\n \\frac{dy}{dt} -y=6 \\exp(t)+42 \\exp(8 t) with $y(0)=7.$ $y=$ [ANS].", "answer_v1": [ "(7 - 6 )* exp(t) + 6 * t * exp(t) + 6 * exp(8 *t)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the initial value problem\n \\frac{dy}{dt} -y=9 \\exp(t)+2 \\exp(2 t) with $y(0)=4.$ $y=$ [ANS].", "answer_v2": [ "(4 - 2 )* exp(t) + 9 * t * exp(t) + 2 * exp(2 *t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the initial value problem\n \\frac{dy}{dt} -y=6 \\exp(t)+9 \\exp(4 t) with $y(0)=5.$ $y=$ [ANS].", "answer_v3": [ "(5 - 3 )* exp(t) + 6 * t * exp(t) + 3 * exp(4 *t)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0167", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Integrating factor", "level": "3", "keywords": [ "calculus", "integral", "differential equations" ], "problem_v1": "(a) Consider the differential equation x \\frac{dy}{dx} -4 y=0. Find a general solution to this differential equation that has the form $y=Cx^n$. $y=$ [ANS]. Find a second solution $y=Cx^n$ that might not be a general solution and which may have a different value of $n$ than your first solution. $y=$ [ANS]. (b) If the solution additionally satisfies $y=35$ when $x=4$, what is the solution? $y=$ [ANS]", "answer_v1": [ "C*x^4", "0*x^n", "35*(x/4)^4" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "(a) Consider the differential equation x \\frac{dy}{dx} -2 y=0. Find a general solution to this differential equation that has the form $y=Cx^n$. $y=$ [ANS]. Find a second solution $y=Cx^n$ that might not be a general solution and which may have a different value of $n$ than your first solution. $y=$ [ANS]. (b) If the solution additionally satisfies $y=15$ when $x=5$, what is the solution? $y=$ [ANS]", "answer_v2": [ "C*x^2", "0*x^n", "15*(x/5)^2" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "(a) Consider the differential equation x \\frac{dy}{dx} -2 y=0. Find a general solution to this differential equation that has the form $y=Cx^n$. $y=$ [ANS]. Find a second solution $y=Cx^n$ that might not be a general solution and which may have a different value of $n$ than your first solution. $y=$ [ANS]. (b) If the solution additionally satisfies $y=20$ when $x=4$, what is the solution? $y=$ [ANS]", "answer_v3": [ "C*x^2", "0*x^n", "20*(x/4)^2" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0168", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Integrating factor", "level": "3", "keywords": [ "differential equation", "calculus", "antiderivatives'\"" ], "problem_v1": "Find the solution of the initial value problem. ${dq\\over dz}=7+\\sin z$, and $q=10$ when $z=0$. $q(z)=$ [ANS]", "answer_v1": [ "7*z-cos(z)+11" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the solution of the initial value problem. ${dq\\over dz}=2+\\sin z$, and $q=7$ when $z=0$. $q(z)=$ [ANS]", "answer_v2": [ "2*z-cos(z)+8" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the solution of the initial value problem. ${dq\\over dz}=4+\\sin z$, and $q=7$ when $z=0$. $q(z)=$ [ANS]", "answer_v3": [ "4*z-cos(z)+8" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0169", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Integrating factor", "level": "2", "keywords": [ "derivative" ], "problem_v1": "Solve the initial value problem. \\frac{dy}{dx} -2x y=10x,\\;y(0)=1 $ y=$ [ANS]", "answer_v1": [ "6*e^(x^2)-5" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the initial value problem. \\frac{dy}{dx} -2x y=4x,\\;y(0)=7 $ y=$ [ANS]", "answer_v2": [ "9*e^(x^2)-2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the initial value problem. \\frac{dy}{dx} -2x y=6x,\\;y(0)=2 $ y=$ [ANS]", "answer_v3": [ "5*e^(x^2)-3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0170", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Integrating factor", "level": "2", "keywords": [ "differential equations", "first order", "integrating factor" ], "problem_v1": "Consider the intial value problem\n \\frac{dy}{dt} +6 y=5, \\ \\ \\ y(0)=0. Find a nonzero solution to the associated homogeneous differential equation. $y=$ [ANS]\nFind a particular solution to the nonhomogeneous differential equation. $y=$ [ANS]\nFind the solution to the initial value problem. $y=$ [ANS]", "answer_v1": [ "e^(-6*t)", "0.833333", "5/6*[1-e^(-6*t)]" ], "answer_type_v1": [ "EX", "NV", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Consider the intial value problem\n \\frac{dy}{dt} +2 y=7, \\ \\ \\ y(0)=0. Find a nonzero solution to the associated homogeneous differential equation. $y=$ [ANS]\nFind a particular solution to the nonhomogeneous differential equation. $y=$ [ANS]\nFind the solution to the initial value problem. $y=$ [ANS]", "answer_v2": [ "e^(-2*t)", "3.5", "7/2*[1-e^(-2*t)]" ], "answer_type_v2": [ "EX", "NV", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Consider the intial value problem\n \\frac{dy}{dt} +3 y=5, \\ \\ \\ y(0)=0. Find a nonzero solution to the associated homogeneous differential equation. $y=$ [ANS]\nFind a particular solution to the nonhomogeneous differential equation. $y=$ [ANS]\nFind the solution to the initial value problem. $y=$ [ANS]", "answer_v3": [ "e^{-3*t}", "1.66667", "5/3*[1-e^{-3*t}]" ], "answer_type_v3": [ "EX", "NV", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0171", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Integrating factor", "level": "3", "keywords": [ "differential equations", "first order", "integrating factor" ], "problem_v1": "Find the general solution to\n(t^2+25) y^{\\,\\prime}+2 t y=t^2 (t^2+25). Enter your answer as $y=\\dots$. Use $C$ to denote the arbitrary constant in your answer. [ANS]", "answer_v1": [ "y = (t^5/5+25*t^3/3+C)/(t^2+25)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the general solution to\n(t^2+4) y^{\\,\\prime}+2 t y=t^2 (t^2+4). Enter your answer as $y=\\dots$. Use $C$ to denote the arbitrary constant in your answer. [ANS]", "answer_v2": [ "y = (t^5/5+4*t^3/3+C)/(t^2+4)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the general solution to\n(t^2+9) y^{\\,\\prime}+2 t y=t^2 (t^2+9). Enter your answer as $y=\\dots$. Use $C$ to denote the arbitrary constant in your answer. [ANS]", "answer_v3": [ "y = (t^5/5+9*t^3/3+C)/(t^2+9)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0172", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Integrating factor", "level": "4", "keywords": [ "differential equations", "first order", "integrating factor" ], "problem_v1": "Find the solution to the initial value problem \\frac{y^{\\,\\prime}-e^{-t}+6}{y} =-6, \\ \\ \\ y(0)=1. [ANS]\nDiscuss the behavior of the solution $y(t)$ as $t$ becomes large. Does $ \\lim_{t \\to \\infty} y(t)$ exist? If the limit exists, enter its value. If the limit does not exist, enter DNE.\n$ \\lim_{t \\to \\infty} y(t)=$ [ANS]", "answer_v1": [ "y = 1/5*e^(-t)-1+9/5*e^(-6*t)", "-1" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the solution to the initial value problem \\frac{y^{\\,\\prime}-e^{-t}+3}{y} =-3, \\ \\ \\ y(0)=5. [ANS]\nDiscuss the behavior of the solution $y(t)$ as $t$ becomes large. Does $ \\lim_{t \\to \\infty} y(t)$ exist? If the limit exists, enter its value. If the limit does not exist, enter DNE.\n$ \\lim_{t \\to \\infty} y(t)=$ [ANS]", "answer_v2": [ "y = 1/2*e^(-t)-1+11/2*e^{-3*t}", "-1" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the solution to the initial value problem \\frac{y^{\\,\\prime}-e^{-t}+4}{y} =-4, \\ \\ \\ y(0)=1. [ANS]\nDiscuss the behavior of the solution $y(t)$ as $t$ becomes large. Does $ \\lim_{t \\to \\infty} y(t)$ exist? If the limit exists, enter its value. If the limit does not exist, enter DNE.\n$ \\lim_{t \\to \\infty} y(t)=$ [ANS]", "answer_v3": [ "y = 1/3*e^(-t)-1+5/3*e^{-4*t}", "-1" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0173", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Integrating factor", "level": "3", "keywords": [ "differential equations", "first order", "integrating factor" ], "problem_v1": "Find the general solution to\nt \\ln(t) \\frac{dr}{dt} +r=8 t e^t. Enter your answer as $r=\\dots$. Use $C$ to denote the arbitrary constant in your answer. [ANS]", "answer_v1": [ "r = (8*e^t+C)/[ln(t)]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the general solution to\nt \\ln(t) \\frac{dr}{dt} +r=2 t e^t. Enter your answer as $r=\\dots$. Use $C$ to denote the arbitrary constant in your answer. [ANS]", "answer_v2": [ "r = (2*e^t+C)/[ln(t)]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the general solution to\nt \\ln(t) \\frac{dr}{dt} +r=4 t e^t. Enter your answer as $r=\\dots$. Use $C$ to denote the arbitrary constant in your answer. [ANS]", "answer_v3": [ "r = (4*e^t+C)/[ln(t)]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0174", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Integrating factor", "level": "3", "keywords": [], "problem_v1": "Solve the initial value problem\n12 (t+1) \\frac{dy}{dt} -8 y=32 t, for $t >-1$ with $y(0)=13.$\nFind the integrating factor, $u(t)=$ [ANS], and then find $y(t)=$ [ANS].", "answer_v1": [ "(t+1)^(-0.666666666666667)", "8*t +12 + (1 * ((t + 1)**0.666666666666667))" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Solve the initial value problem\n6 (t+1) \\frac{dy}{dt} -5 y=5 t, for $t >-1$ with $y(0)=19.$\nFind the integrating factor, $u(t)=$ [ANS], and then find $y(t)=$ [ANS].", "answer_v2": [ "(t+1)^(-0.833333333333333)", "5*t +6 + (13 * ((t + 1)**0.833333333333333))" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Solve the initial value problem\n8 (t+1) \\frac{dy}{dt} -6 y=12 t, for $t >-1$ with $y(0)=13.$\nFind the integrating factor, $u(t)=$ [ANS], and then find $y(t)=$ [ANS].", "answer_v3": [ "(t+1)^(-0.75)", "6*t +8 + (5 * ((t + 1)**0.75))" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0175", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Integrating factor", "level": "3", "keywords": [], "problem_v1": "Find the general solution to the differential equation x^2+2x y+x \\frac{dy}{dx} =0\nFind the integrating factor, $u(x)=$ [ANS].\nFind $y(x)=$ [ANS]. Use C as the unknown constant.", "answer_v1": [ "e^(2 x)", "C e^(-2 x) - x/2 + 1/2^2 " ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Find the general solution to the differential equation x^2-4x y+x \\frac{dy}{dx} =0\nFind the integrating factor, $u(x)=$ [ANS].\nFind $y(x)=$ [ANS]. Use C as the unknown constant.", "answer_v2": [ "e^(-4 x)", "C e^(4 x) + x/4 + 1/4^2 " ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Find the general solution to the differential equation x^2-2x y+x \\frac{dy}{dx} =0\nFind the integrating factor, $u(x)=$ [ANS].\nFind $y(x)=$ [ANS]. Use C as the unknown constant.", "answer_v3": [ "e^(-2 x)", "C e^(2 x) + x/2 + 1/2^2 " ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0176", "subject": "Differential_equations", "topic": "First order differential equations", "subtopic": "Integrating factor", "level": "3", "keywords": [ "differential equations", "separable", "differential equation' 'linear", "Differential equations", "Linear 1st order" ], "problem_v1": "Solve the initial value problem\n \\frac{dy}{dt} +2y=40 \\sin(t)+30 \\cos(t) with $y(0)=6.$ $y=$ [ANS]. Reminder: To find the anti-derivative of $e^u\\sin(u)$, the trick is to do integration by parts twice.", "answer_v1": [ "(2*8 + 6 )* sin(t) + (2*6 - 8 )* cos(t) + 2 * exp(-2* t)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the initial value problem\n \\frac{dy}{dt} +2y=10 \\sin(t)+45 \\cos(t) with $y(0)=2.$ $y=$ [ANS]. Reminder: To find the anti-derivative of $e^u\\sin(u)$, the trick is to do integration by parts twice.", "answer_v2": [ "(2*2 + 9 )* sin(t) + (2*9 - 2 )* cos(t) - 14 * exp(-2* t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the initial value problem\n \\frac{dy}{dt} +2y=20 \\sin(t)+30 \\cos(t) with $y(0)=3.$ $y=$ [ANS]. Reminder: To find the anti-derivative of $e^u\\sin(u)$, the trick is to do integration by parts twice.", "answer_v3": [ "(2*4 + 6 )* sin(t) + (2*6 - 4 )* cos(t) - 5 * exp(-2* t)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0177", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Reduction of order", "level": "3", "keywords": [ "Differential Equation", "Solution", "Reduction" ], "problem_v1": "We know that $y_1(x)=e^{8x}\\sin(16x)$ is a solution to the differential equation $D^2y-16 D y+320 y=0$ for $x \\in (-\\infty,\\infty)$. Use the method of reduction of order to find a second solution to $D^2y-16 D y+320 y=0$ for $x \\in (-\\infty,\\infty)$.\n(a) After you reduce the second order equation by making the substitution $w=u'$, you get a first order equation of the form $w'=f(x,w)=$-3*R1+R3-3*R1+R3. Note: Make sure you use a lower case w, (and don't use w(t), it confuses the computer).\n(b) A second solution to $D^2y-16 D y+320 y=0$ for $x \\in (-\\infty,\\infty)$ is $y_2(x)=$ [ANS]", "answer_v1": [ "-2 * 16 * cot(16 x)*w", "c*e^(8 x)*cos(16 x)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "We know that $y_1(x)=e^{2x}\\sin(19x)$ is a solution to the differential equation $D^2y-4 D y+365 y=0$ for $x \\in (-\\infty,\\infty)$. Use the method of reduction of order to find a second solution to $D^2y-4 D y+365 y=0$ for $x \\in (-\\infty,\\infty)$.\n(a) After you reduce the second order equation by making the substitution $w=u'$, you get a first order equation of the form $w'=f(x,w)=$-3*R1+R3-3*R1+R3. Note: Make sure you use a lower case w, (and don't use w(t), it confuses the computer).\n(b) A second solution to $D^2y-4 D y+365 y=0$ for $x \\in (-\\infty,\\infty)$ is $y_2(x)=$ [ANS]", "answer_v2": [ "-2 * 19 * cot(19 x)*w", "c*e^(2 x)*cos(19 x)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "We know that $y_1(x)=e^{4x}\\sin(16x)$ is a solution to the differential equation $D^2y-8 D y+272 y=0$ for $x \\in (-\\infty,\\infty)$. Use the method of reduction of order to find a second solution to $D^2y-8 D y+272 y=0$ for $x \\in (-\\infty,\\infty)$.\n(a) After you reduce the second order equation by making the substitution $w=u'$, you get a first order equation of the form $w'=f(x,w)=$-3*R1+R3-3*R1+R3. Note: Make sure you use a lower case w, (and don't use w(t), it confuses the computer).\n(b) A second solution to $D^2y-8 D y+272 y=0$ for $x \\in (-\\infty,\\infty)$ is $y_2(x)=$ [ANS]", "answer_v3": [ "-2 * 16 * cot(16 x)*w", "c*e^(4 x)*cos(16 x)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0178", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Reduction of order", "level": "4", "keywords": [], "problem_v1": "Consider the DE \\frac{d^2y}{dx^2} +10 \\frac{dy}{dx} +25y=x which is linear with constant coefficients.\nFirst we will work on solving the corresponding homogeneous equation. The auxiliary equation (using m as your variable) is [ANS] $=0$ which has root [ANS].\nBecause this is a repeated root, we don't have much choice but to use the exponential function corresponding to this root: [ANS] to do reduction of order. $y_2=u e^{-5x}$. Then (using the prime notation for the derivatives) $y_2^\\prime=$ [ANS]\n$y_2^{\\prime\\prime}=$ [ANS]\nSo, plugging $y_2$ into the left side of the differential equation, and reducing, we get\n$y_2^{\\prime\\prime}+10 y_2^\\prime+25 y_2=$ [ANS]\nSo now our equation is $e^{-5x} u^{\\prime\\prime}=x$. To solve for u we need only integrate $x e^{5x}$ twice, using a as our first constant of integration and b as the second we get $u=$ [ANS]\nTherefore $y_2=$ [ANS], the general solution.\nWe knew from the beginning that $e^{-5x}$ was a solution. We have worked out is that $x e^{-5x}$ is another solution to the homogeneous equation, which is generally the case when we have multiple roots. Then $ \\frac{5x-2}{125} $ is the particular solution to the nonhomogeneous equation, and the general solution we derived is pieced together using superposition.", "answer_v1": [ "m^2+10*m+25", "-5", "e^(-5*x)", "e^(-5*x)*(-5*u+u", "e^(-5*x)*(25*u-10*u", "e^(-5*x)*u", "(-2+5*x)*e^(5*x)/125+a*x+b", "(5*x-2)/125+(a*x+b)*e^(-5*x)" ], "answer_type_v1": [ "EX", "NV", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the DE \\frac{d^2y}{dx^2} -16 \\frac{dy}{dx} +64y=x which is linear with constant coefficients.\nFirst we will work on solving the corresponding homogeneous equation. The auxiliary equation (using m as your variable) is [ANS] $=0$ which has root [ANS].\nBecause this is a repeated root, we don't have much choice but to use the exponential function corresponding to this root: [ANS] to do reduction of order. $y_2=u e^{8x}$. Then (using the prime notation for the derivatives) $y_2^\\prime=$ [ANS]\n$y_2^{\\prime\\prime}=$ [ANS]\nSo, plugging $y_2$ into the left side of the differential equation, and reducing, we get\n$y_2^{\\prime\\prime}-16 y_2^\\prime+64 y_2=$ [ANS]\nSo now our equation is $e^{8x} u^{\\prime\\prime}=x$. To solve for u we need only integrate $x e^{-8x}$ twice, using a as our first constant of integration and b as the second we get $u=$ [ANS]\nTherefore $y_2=$ [ANS], the general solution.\nWe knew from the beginning that $e^{8x}$ was a solution. We have worked out is that $x e^{8x}$ is another solution to the homogeneous equation, which is generally the case when we have multiple roots. Then $ \\frac{2+8x}{512} $ is the particular solution to the nonhomogeneous equation, and the general solution we derived is pieced together using superposition.", "answer_v2": [ "m^2-16*m+64", "8", "e^(8*x)", "e^(8*x)*(8*u+u", "e^(8*x)*[64*u-(-16)*u", "e^(8*x)*u", "[-2+(-8)*x]*e^(-8*x)/(-512)+a*x+b", "(2+8*x)/512+(a*x+b)*e^(8*x)" ], "answer_type_v2": [ "EX", "NV", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the DE \\frac{d^2y}{dx^2} -8 \\frac{dy}{dx} +16y=x which is linear with constant coefficients.\nFirst we will work on solving the corresponding homogeneous equation. The auxiliary equation (using m as your variable) is [ANS] $=0$ which has root [ANS].\nBecause this is a repeated root, we don't have much choice but to use the exponential function corresponding to this root: [ANS] to do reduction of order. $y_2=u e^{4x}$. Then (using the prime notation for the derivatives) $y_2^\\prime=$ [ANS]\n$y_2^{\\prime\\prime}=$ [ANS]\nSo, plugging $y_2$ into the left side of the differential equation, and reducing, we get\n$y_2^{\\prime\\prime}-8 y_2^\\prime+16 y_2=$ [ANS]\nSo now our equation is $e^{4x} u^{\\prime\\prime}=x$. To solve for u we need only integrate $x e^{-4x}$ twice, using a as our first constant of integration and b as the second we get $u=$ [ANS]\nTherefore $y_2=$ [ANS], the general solution.\nWe knew from the beginning that $e^{4x}$ was a solution. We have worked out is that $x e^{4x}$ is another solution to the homogeneous equation, which is generally the case when we have multiple roots. Then $ \\frac{2+4x}{64} $ is the particular solution to the nonhomogeneous equation, and the general solution we derived is pieced together using superposition.", "answer_v3": [ "m^2-8*m+16", "4", "e^(4*x)", "e^(4*x)*(4*u+u", "e^(4*x)*[16*u-(-8)*u", "e^(4*x)*u", "[-2+(-4)*x]*e^(-4*x)/(-64)+a*x+b", "(2+4*x)/64+(a*x+b)*e^(4*x)" ], "answer_type_v3": [ "EX", "NV", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Differential_equations_0180", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Applications", "level": "5", "keywords": [ "differential equation' 'second order' 'linear' 'nonhomogeneous" ], "problem_v1": "This problem is an example of critically damped harmonic motion. A hollow steel ball weighing 4 pounds is suspended from a spring. This stretches the spring $ \\frac{1}{8} $ feet.\nThe ball is started in motion from the equilibrium position with a downward velocity of $8$ feet per second. The air resistance (in pounds) of the moving ball numerically equals 4 times its velocity (in feet per second). Suppose that after t seconds the ball is y feet below its rest position. Find y in terms of t. Take as the gravitational acceleration 32 feet per second. (Note that the positive y direction is down in this problem.)\n$y=$-3*R1+R3-3*R1+R3", "answer_v1": [ "(0) *exp((-16)*t) + (8)*t *exp((-16)*t)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "This problem is an example of critically damped harmonic motion. A hollow steel ball weighing 4 pounds is suspended from a spring. This stretches the spring $ \\frac{1}{8} $ feet.\nThe ball is started in motion from the equilibrium position with a downward velocity of $2$ feet per second. The air resistance (in pounds) of the moving ball numerically equals 4 times its velocity (in feet per second). Suppose that after t seconds the ball is y feet below its rest position. Find y in terms of t. Take as the gravitational acceleration 32 feet per second. (Note that the positive y direction is down in this problem.)\n$y=$-3*R1+R3-3*R1+R3", "answer_v2": [ "(0) *exp((-16)*t) + (2)*t *exp((-16)*t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "This problem is an example of critically damped harmonic motion. A hollow steel ball weighing 4 pounds is suspended from a spring. This stretches the spring $ \\frac{1}{8} $ feet.\nThe ball is started in motion from the equilibrium position with a downward velocity of $4$ feet per second. The air resistance (in pounds) of the moving ball numerically equals 4 times its velocity (in feet per second). Suppose that after t seconds the ball is y feet below its rest position. Find y in terms of t. Take as the gravitational acceleration 32 feet per second. (Note that the positive y direction is down in this problem.)\n$y=$-3*R1+R3-3*R1+R3", "answer_v3": [ "(0) *exp((-16)*t) + (4)*t *exp((-16)*t)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0181", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Applications", "level": "5", "keywords": [], "problem_v1": "A cylindrical tank with height $8$, radius $3$, and weight $35$ lbs is vertically submerged in water, with y(t) the height of the tank above the water. Two forces act on the tank, the acceleration due to gravity, $32 \\frac{ft}{s^2} $ acting in the opposite direction of y and the buoyancy force by Archimedes principle. If water weighs $62.4 \\frac{lbs}{ft^3} $ then the buoyancy force is [ANS]\nTherefore the differential equation modeling the forces on the tank is [ANS] $=0$\nNote that weight=force=ma", "answer_v1": [ "62.4*(8-y)*9*pi", "35*y" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "A cylindrical tank with height $10$, radius $2$, and weight $21$ lbs is vertically submerged in water, with y(t) the height of the tank above the water. Two forces act on the tank, the acceleration due to gravity, $32 \\frac{ft}{s^2} $ acting in the opposite direction of y and the buoyancy force by Archimedes principle. If water weighs $62.4 \\frac{lbs}{ft^3} $ then the buoyancy force is [ANS]\nTherefore the differential equation modeling the forces on the tank is [ANS] $=0$\nNote that weight=force=ma", "answer_v2": [ "62.4*(10-y)*4*pi", "21*y" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "A cylindrical tank with height $8$, radius $2$, and weight $26$ lbs is vertically submerged in water, with y(t) the height of the tank above the water. Two forces act on the tank, the acceleration due to gravity, $32 \\frac{ft}{s^2} $ acting in the opposite direction of y and the buoyancy force by Archimedes principle. If water weighs $62.4 \\frac{lbs}{ft^3} $ then the buoyancy force is [ANS]\nTherefore the differential equation modeling the forces on the tank is [ANS] $=0$\nNote that weight=force=ma", "answer_v3": [ "62.4*(8-y)*4*pi", "26*y" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0182", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Applications", "level": "5", "keywords": [], "problem_v1": "An object with mass $22$ kg is in free fall. It has two forces acting on it: gravity and air resistance. For this model we will take the ground to be at height 0, the height of the object to be y(t) meters from the ground, and the acceleration due to gravity to be $-9.81 \\frac{m}{s^2} $. We want to set up a differential equation for the forces acting on the object. The force due to gravity is given by mass times acceleration acting in the opposite direction of y. The force due to air resistance is proportional to the square root of the velocity of the object in the same direction is y. The constant of proportionality for the wind resistance has been measured at $0.55$. Find the differential equation that models the displacement of the object y at time t. [ANS] $=0$ If the object is dropped from a height of $16$ the IVP is $y(0)=$ [ANS]\n$y^\\prime (0)=$ [ANS]\nNote: use y,y', etc instead of y(t), y'(t) in your answers. After you set up your differential equation you will have to set it equal to zero so that WeBWorK will understand your answer, do this in a way so that the highest order derivative has a positive coefficient.", "answer_v1": [ "22*y", "16", "0" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "An object with mass $29$ kg is in free fall. It has two forces acting on it: gravity and air resistance. For this model we will take the ground to be at height 0, the height of the object to be y(t) meters from the ground, and the acceleration due to gravity to be $-9.81 \\frac{m}{s^2} $. We want to set up a differential equation for the forces acting on the object. The force due to gravity is given by mass times acceleration acting in the opposite direction of y. The force due to air resistance is proportional to the square root of the velocity of the object in the same direction is y. The constant of proportionality for the wind resistance has been measured at $0.1$. Find the differential equation that models the displacement of the object y at time t. [ANS] $=0$ If the object is dropped from a height of $11$ the IVP is $y(0)=$ [ANS]\n$y^\\prime (0)=$ [ANS]\nNote: use y,y', etc instead of y(t), y'(t) in your answers. After you set up your differential equation you will have to set it equal to zero so that WeBWorK will understand your answer, do this in a way so that the highest order derivative has a positive coefficient.", "answer_v2": [ "29*y", "11", "0" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "An object with mass $22$ kg is in free fall. It has two forces acting on it: gravity and air resistance. For this model we will take the ground to be at height 0, the height of the object to be y(t) meters from the ground, and the acceleration due to gravity to be $-9.81 \\frac{m}{s^2} $. We want to set up a differential equation for the forces acting on the object. The force due to gravity is given by mass times acceleration acting in the opposite direction of y. The force due to air resistance is proportional to the square root of the velocity of the object in the same direction is y. The constant of proportionality for the wind resistance has been measured at $0.25$. Find the differential equation that models the displacement of the object y at time t. [ANS] $=0$ If the object is dropped from a height of $13$ the IVP is $y(0)=$ [ANS]\n$y^\\prime (0)=$ [ANS]\nNote: use y,y', etc instead of y(t), y'(t) in your answers. After you set up your differential equation you will have to set it equal to zero so that WeBWorK will understand your answer, do this in a way so that the highest order derivative has a positive coefficient.", "answer_v3": [ "22*y", "13", "0" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0183", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Applications", "level": "3", "keywords": [ "hooke's", "law", "damping" ], "problem_v1": "If the differential equation m \\frac{d^2x}{dt^2} +8 \\frac{dx}{dt} +6x=0 is overdamped, the range of values for m is? [ANS]\nYour answer will be an interval of numbers given in the form (1,2), [1,2), (-inf,6], etc.", "answer_v1": [ "(0,8^2/(4*6))" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "If the differential equation m \\frac{d^2x}{dt^2} +2 \\frac{dx}{dt} +9x=0 is overdamped, the range of values for m is? [ANS]\nYour answer will be an interval of numbers given in the form (1,2), [1,2), (-inf,6], etc.", "answer_v2": [ "(0,2^2/(4*9))" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "If the differential equation m \\frac{d^2x}{dt^2} +4 \\frac{dx}{dt} +6x=0 is overdamped, the range of values for m is? [ANS]\nYour answer will be an interval of numbers given in the form (1,2), [1,2), (-inf,6], etc.", "answer_v3": [ "(0,4^2/(4*6))" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0184", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Applications", "level": "3", "keywords": [ "hooke's", "law", "damping" ], "problem_v1": "Solve the differential equation for the motion of the spring 1728 \\frac{d^2x}{dt^2} +4608 \\frac{dx}{dt} +3072x=0 if $x(0)=2$ $ \\frac{dx}{dt} \\Big\\vert_{t=0}=4$\n$x(t)=$ [ANS]\nThis spring is critically damped, will it go past equilibrium? [ANS]", "answer_v1": [ "e^(-1.33333*t)*(2+6.66667*t)", "N" ], "answer_type_v1": [ "EX", "TF" ], "options_v1": [ [], [] ], "problem_v2": "Solve the differential equation for the motion of the spring 1458 \\frac{d^2x}{dt^2} +648 \\frac{dx}{dt} +72x=0 if $x(0)=-7$ $ \\frac{dx}{dt} \\Big\\vert_{t=0}=-3$\n$x(t)=$ [ANS]\nThis spring is critically damped, will it go past equilibrium? [ANS]", "answer_v2": [ "e^(-0.222222*t)*[-7+(-4.55556)*t]", "N" ], "answer_type_v2": [ "EX", "TF" ], "options_v2": [ [], [] ], "problem_v3": "Solve the differential equation for the motion of the spring 864 \\frac{d^2x}{dt^2} +1152 \\frac{dx}{dt} +384x=0 if $x(0)=-4$ $ \\frac{dx}{dt} \\Big\\vert_{t=0}=1$\n$x(t)=$ [ANS]\nThis spring is critically damped, will it go past equilibrium? [ANS]", "answer_v3": [ "e^(-0.666667*t)*[-4+(-1.66667)*t]", "N" ], "answer_type_v3": [ "EX", "TF" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0185", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Applications", "level": "3", "keywords": [ "calculus", "differential equations", "trigonometric functions" ], "problem_v1": "Find the amplitude of $6 \\sin 4 t+5\\cos 4 t$. amplitude=[ANS]", "answer_v1": [ "sqrt(6^2+5^2)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the amplitude of $2 \\sin 2 t+7\\cos 2 t$. amplitude=[ANS]", "answer_v2": [ "sqrt(2^2+7^2)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the amplitude of $3 \\sin 3 t+5\\cos 3 t$. amplitude=[ANS]", "answer_v3": [ "sqrt(3^2+5^2)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0186", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Applications", "level": "5", "keywords": [ "calculus", "differential equations", "trigonometric functions" ], "problem_v1": "A brick of mass 8 kg hangs from the end of a spring. When the brick is at rest, the spring is stretched by 7 cm. The spring is then stretched an additional 3 cm and released. Assume there is no air resistance. Note that the acceleration due to gravity, $g$, is $g=980$ cm/s ${}^2$. Set up a differential equation with initial conditions describing the motion and solve it for the displacement $s(t)$ of the mass from its equilibrium position (with the spring stretched 7 cm). $s(t)=$ [ANS] cm (Note that your answer should measure t in seconds and s in centimeters.) (Note that your answer should measure t in seconds and s in centimeters.)", "answer_v1": [ "3*cos(sqrt(980/7)*t)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "A brick of mass 2 kg hangs from the end of a spring. When the brick is at rest, the spring is stretched by 9 cm. The spring is then stretched an additional 2 cm and released. Assume there is no air resistance. Note that the acceleration due to gravity, $g$, is $g=980$ cm/s ${}^2$. Set up a differential equation with initial conditions describing the motion and solve it for the displacement $s(t)$ of the mass from its equilibrium position (with the spring stretched 9 cm). $s(t)=$ [ANS] cm (Note that your answer should measure t in seconds and s in centimeters.) (Note that your answer should measure t in seconds and s in centimeters.)", "answer_v2": [ "2*cos(sqrt(980/9)*t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "A brick of mass 4 kg hangs from the end of a spring. When the brick is at rest, the spring is stretched by 7 cm. The spring is then stretched an additional 2 cm and released. Assume there is no air resistance. Note that the acceleration due to gravity, $g$, is $g=980$ cm/s ${}^2$. Set up a differential equation with initial conditions describing the motion and solve it for the displacement $s(t)$ of the mass from its equilibrium position (with the spring stretched 7 cm). $s(t)=$ [ANS] cm (Note that your answer should measure t in seconds and s in centimeters.) (Note that your answer should measure t in seconds and s in centimeters.)", "answer_v3": [ "2*cos(sqrt(980/7)*t)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0187", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Applications", "level": "5", "keywords": [ "calculus", "integral", "differential equations", "higher derivatives" ], "problem_v1": "For the differential equation s''+b s'+8 s=0, find the values of $b$ that make the general solution overdamped, underdamped, or critically damped.\n(For each, give an interval or intervals for b for which the equation is as indicated. Thus if the equation is overdamped for all b in the range $-1 0$).\n y''+{\\lambda}y=0 \\qquad \\mathrm{with} \\quad y'(0)=0, \\quad y'(5)=0. Eigenvalues: $\\quad \\,\\,\\lambda_n=$ [ANS]\nEigenfunctions: $\\ \\ y_n=$ [ANS]\nNotation: Your answers should involve $n$ and $x$.\nIf you don't get this in 2 tries, you can get a hint.", "answer_v1": [ "(n*pi/5)^2", "cos(n*pi/5*x)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Find the eigenvalues and eigenfunctions for the following boundary value problem (with $\\lambda > 0$).\n y''+{\\lambda}y=0 \\qquad \\mathrm{with} \\quad y'(0)=0, \\quad y(1)=0. Eigenvalues: $\\quad \\,\\,\\lambda_n=$ [ANS]\nEigenfunctions: $\\ \\ y_n=$ [ANS]\nNotation: Your answers should involve $n$ and $x$.\nIf you don't get this in 2 tries, you can get a hint.", "answer_v2": [ "[(2*n+1)*pi/2]^2", "cos((2*n+1)*pi/2*x)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Find the eigenvalues and eigenfunctions for the following boundary value problem (with $\\lambda > 0$).\n y''+{\\lambda}y=0 \\qquad \\mathrm{with} \\quad y'(0)=0, \\quad y(2)=0. Eigenvalues: $\\quad \\,\\,\\lambda_n=$ [ANS]\nEigenfunctions: $\\ \\ y_n=$ [ANS]\nNotation: Your answers should involve $n$ and $x$.\nIf you don't get this in 2 tries, you can get a hint.", "answer_v3": [ "[(2*n+1)*pi/4]^2", "cos((2*n+1)*pi/4*x)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0218", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Boundary value problems", "level": "3", "keywords": [ "ODE", "boundary value" ], "problem_v1": "Solve the following differential equation with the given boundary conditions.-If there are infinitely many solutions, use c for any undetermined constants.-If there are no solutions, write No Solution.-Write answers as functions of $x$ (i.e. $y=y(x)$).\n y''+16y=0\nA) Boundary conditions: $ \\quad \\begin{aligned}[t] y(0) &=-1 \\\\ y\\left( \\frac{7\\pi}{4} \\right) &=-1 \\end{aligned}$ $y=$ [ANS]\nB) Boundary conditions: $ \\quad \\begin{aligned}[t] y(0) &=-1 \\\\ y\\left( \\frac{15\\pi}{8} \\right) &=1 \\end{aligned}$ $y=$ [ANS]\nC) Boundary conditions: $ \\quad \\begin{aligned}[t] y(0) &=-1 \\\\ y\\left( \\frac{3\\pi}{2} \\right) &=-1 \\end{aligned}$ $y=$ [ANS]", "answer_v1": [ "DNE", "-1*cos(4*x)+(-1)*sin(4*x)", "-1*cos(4*x)+c*sin(4*x)" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Solve the following differential equation with the given boundary conditions.-If there are infinitely many solutions, use c for any undetermined constants.-If there are no solutions, write No Solution.-Write answers as functions of $x$ (i.e. $y=y(x)$).\n y''+y=0\nA) Boundary conditions: $ \\quad \\begin{aligned}[t] y(0) &=3 \\\\ y\\left(3\\pi \\right) &=-3 \\end{aligned}$ $y=$ [ANS]\nB) Boundary conditions: $ \\quad \\begin{aligned}[t] y(0) &=3 \\\\ y\\left(\\pi \\right) &=3 \\end{aligned}$ $y=$ [ANS]\nC) Boundary conditions: $ \\quad \\begin{aligned}[t] y(0) &=3 \\\\ y\\left( \\frac{3\\pi}{2} \\right) &=-3 \\end{aligned}$ $y=$ [ANS]", "answer_v2": [ "3*cos(1*x)+c*sin(1*x)", "DNE", "3*cos(1*x)+3*sin(1*x)" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Solve the following differential equation with the given boundary conditions.-If there are infinitely many solutions, use c for any undetermined constants.-If there are no solutions, write No Solution.-Write answers as functions of $x$ (i.e. $y=y(x)$).\n y''+4y=0\nA) Boundary conditions: $ \\quad \\begin{aligned}[t] y(0) &=-2 \\\\ y\\left( \\frac{7\\pi}{4} \\right) &=-2 \\end{aligned}$ $y=$ [ANS]\nB) Boundary conditions: $ \\quad \\begin{aligned}[t] y(0) &=-2 \\\\ y\\left(\\pi \\right) &=2 \\end{aligned}$ $y=$ [ANS]\nC) Boundary conditions: $ \\quad \\begin{aligned}[t] y(0) &=-2 \\\\ y\\left(2\\pi \\right) &=-2 \\end{aligned}$ $y=$ [ANS]", "answer_v3": [ "-2*cos(2*x)+2*sin(2*x)", "DNE", "-2*cos(2*x)+c*sin(2*x)" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0219", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Linear independence", "level": "3", "keywords": [ "differential equation' 'second order' 'linear' 'wronskian" ], "problem_v1": "Determine whether the following pairs of functions are linearly independent or not.\n[ANS] 1. $f(\\theta)=12\\cos 3\\theta$ and $g(\\theta)=48\\cos^3\\theta-36\\cos \\theta$ [ANS] 2. $f(t)=t$ and $g(t)=|t|$ [ANS] 3. $f(x)=e^{12x}$ and $g(x)=e^{12(x-1)}$", "answer_v1": [ "LINEARLY DEPENDENT", "LINEARLY INDEPENDENT", "LINEARLY DEPENDENT" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "Linearly dependent" ], [ "Linearly dependent" ], [ "Linearly dependent" ] ], "problem_v2": "Determine whether the following pairs of functions are linearly independent or not.\n[ANS] 1. $f(t)=t$ and $g(t)=|t|$ [ANS] 2. $f(t)=t^2+19t$ and $g(t)=t^2-19t$ [ANS] 3. $f(x)=e^{19x}$ and $g(x)=e^{19(x-1)}$", "answer_v2": [ "LINEARLY INDEPENDENT", "LINEARLY INDEPENDENT", "LINEARLY DEPENDENT" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "Linearly dependent", "LINEARLY INDEPENDENT" ], [ "Linearly dependent", "LINEARLY INDEPENDENT" ], [ "Linearly dependent", "LINEARLY INDEPENDENT" ] ], "problem_v3": "Determine whether the following pairs of functions are linearly independent or not.\n[ANS] 1. $f(t)=t^2+13t$ and $g(t)=t^2-13t$ [ANS] 2. $f(\\theta)=13\\cos 3\\theta$ and $g(\\theta)=52\\cos^3\\theta-39\\cos \\theta$ [ANS] 3. $f(x)=e^{13x}$ and $g(x)=e^{13(x-1)}$", "answer_v3": [ "LINEARLY INDEPENDENT", "LINEARLY DEPENDENT", "LINEARLY DEPENDENT" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "Linearly dependent", "LINEARLY INDEPENDENT" ], [ "Linearly dependent", "LINEARLY INDEPENDENT" ], [ "Linearly dependent", "LINEARLY INDEPENDENT" ] ] }, { "id": "Differential_equations_0220", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Linear independence", "level": "2", "keywords": [], "problem_v1": "It can be shown that $y_1=e^{3x}$ and $y_2=e^{-8x}$ are solutions to the differential equation $D^2 y+5 Dy-24 y=0$ on $(-\\infty,\\infty)$. What does the Wronskian of $y_1, y_2$ equal on $(-\\infty,\\infty)$? $W(y_1,y_2)$=[ANS] on $(-\\infty,\\infty)$.\n[ANS] 1. Is $\\{y_1,y_2\\}$ a fundamental set for $D^2 y+5 Dy-24 y=0$ on $(-\\infty,\\infty)$?", "answer_v1": [ "-[e^{3*x}*8*e^{-8*x}*ln(e)+e^{-8*x}*3*e^{3*x}*ln(e)]", "YES" ], "answer_type_v1": [ "EX", "TF" ], "options_v1": [ [], [ "No" ] ], "problem_v2": "It can be shown that $y_1=e^{2x}$ and $y_2=e^{-9x}$ are solutions to the differential equation $D^2 y+7 Dy-18 y=0$ on $(-\\infty,\\infty)$. What does the Wronskian of $y_1, y_2$ equal on $(-\\infty,\\infty)$? $W(y_1,y_2)$=[ANS] on $(-\\infty,\\infty)$.\n[ANS] 1. Is $\\{y_1,y_2\\}$ a fundamental set for $D^2 y+7 Dy-18 y=0$ on $(-\\infty,\\infty)$?", "answer_v2": [ "-[e^{2*x}*9*e^{-9*x}*ln(e)+e^{-9*x}*2*e^{2*x}*ln(e)]", "YES" ], "answer_type_v2": [ "EX", "TF" ], "options_v2": [ [], [ "No" ] ], "problem_v3": "It can be shown that $y_1=e^{2x}$ and $y_2=e^{-7x}$ are solutions to the differential equation $D^2 y+5 Dy-14 y=0$ on $(-\\infty,\\infty)$. What does the Wronskian of $y_1, y_2$ equal on $(-\\infty,\\infty)$? $W(y_1,y_2)$=[ANS] on $(-\\infty,\\infty)$.\n[ANS] 1. Is $\\{y_1,y_2\\}$ a fundamental set for $D^2 y+5 Dy-14 y=0$ on $(-\\infty,\\infty)$?", "answer_v3": [ "-[e^{2*x}*7*e^{-7*x}*ln(e)+e^{-7*x}*2*e^{2*x}*ln(e)]", "YES" ], "answer_type_v3": [ "EX", "TF" ], "options_v3": [ [], [ "No" ] ] }, { "id": "Differential_equations_0221", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Linear independence", "level": "3", "keywords": [], "problem_v1": "It can be shown that $y_1=x^{-4},\\ y_2=x^{-7}$ and $y_3=6$ are solutions to the differential equation $x^2D^3y+14x D^2y+40 Dy=0$. $W(y_1, y_2, y_3)$=[ANS]. For an IVP with initial conditions at $x=7$, $c_1y_1+c_2y_2+c_3y_3$ is the general solution for x on what interval? [ANS]", "answer_v1": [ "x^(-1-5-8)*(5-8)*(8-1)*(5-1)*6", "(0,infinity)" ], "answer_type_v1": [ "EX", "INT" ], "options_v1": [ [], [] ], "problem_v2": "It can be shown that $y_1=x^{-1},\\ y_2=x^{-8}$ and $y_3=3$ are solutions to the differential equation $x^2D^3y+12x D^2y+18 Dy=0$. $W(y_1, y_2, y_3)$=[ANS]. For an IVP with initial conditions at $x=4$, $c_1y_1+c_2y_2+c_3y_3$ is the general solution for x on what interval? [ANS]", "answer_v2": [ "x^(-1-2-9)*(2-9)*(9-1)*(2-1)*3", "(0,infinity)" ], "answer_type_v2": [ "EX", "INT" ], "options_v2": [ [], [] ], "problem_v3": "It can be shown that $y_1=x^{-2},\\ y_2=x^{-7}$ and $y_3=4$ are solutions to the differential equation $x^2D^3y+12x D^2y+24 Dy=0$. $W(y_1, y_2, y_3)$=[ANS]. For an IVP with initial conditions at $x=6$, $c_1y_1+c_2y_2+c_3y_3$ is the general solution for x on what interval? [ANS]", "answer_v3": [ "x^(-1-3-8)*(3-8)*(8-1)*(3-1)*4", "(0,infinity)" ], "answer_type_v3": [ "EX", "INT" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0222", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Differential operators", "level": "3", "keywords": [ "differential", "equation", "annihilator' 'constant' 'coefficient" ], "problem_v1": "The function $x^{3}e^{5x}\\sin\\!\\left(6x\\right)$ is annihilated by the operator [ANS].", "answer_v1": [ "(D^2-10*D+61)^4" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The function $xe^{-8x}\\sin\\!\\left(9x\\right)$ is annihilated by the operator [ANS].", "answer_v2": [ "[D^2-(-16)*D+145]^2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The function $x^{2}e^{-4x}\\sin\\!\\left(6x\\right)$ is annihilated by the operator [ANS].", "answer_v3": [ "[D^2-(-8)*D+52]^3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0223", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Differential operators", "level": "3", "keywords": [ "differential", "equation", "annihilator' 'constant' 'coefficient" ], "problem_v1": "The differential operator $D^2-10 D+61$ has the form $D^2-2\\alpha D+\\alpha^2+\\beta^2$ where $\\alpha=$ [ANS] and $\\beta=$ [ANS].\nTherefore $D^2-10 D+61$ should annihilate the function $f=e^{5x}\\cos\\!\\left(6x\\right)$ and $g=e^{5x}\\sin\\!\\left(6x\\right)$. We will check that for $e^{5x}\\cos\\!\\left(6x\\right)$. Note that when we compute the derivative and second derivative we will get terms that have $e^{5x}\\sin\\!\\left(6x\\right)$ in them, so we will have to account for them below.\nCompute $D^2\\left(e^{5x}\\cos\\!\\left(6x\\right)\\right)$, $-10 D\\left(e^{5x}\\cos\\!\\left(6x\\right)\\right)$, and $61 e^{5x}\\cos\\!\\left(6x\\right)$. Place the coefficients from the terms $e^{5x}\\cos(6x)$ and $e^{5x}\\sin(6x)$ in the table. The columns of the table should add to zero.\n$\\begin{array}{ccc}\\hline & e^{5x}\\cos\\!\\left(6x\\right) & e^{5x}\\sin\\!\\left(6x\\right) \\\\ \\hline 61 & [ANS] & [ANS] \\\\ \\hline-10 D & [ANS] & [ANS] \\\\ \\hline D^2 & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "5", "6", "61", "0", "-50", "60", "-11", "-60" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "The differential operator $D^2+16 D+145$ has the form $D^2-2\\alpha D+\\alpha^2+\\beta^2$ where $\\alpha=$ [ANS] and $\\beta=$ [ANS].\nTherefore $D^2+16 D+145$ should annihilate the function $f=e^{-8x}\\cos\\!\\left(9x\\right)$ and $g=e^{-8x}\\sin\\!\\left(9x\\right)$. We will check that for $e^{-8x}\\cos\\!\\left(9x\\right)$. Note that when we compute the derivative and second derivative we will get terms that have $e^{-8x}\\sin\\!\\left(9x\\right)$ in them, so we will have to account for them below.\nCompute $D^2\\left(e^{-8x}\\cos\\!\\left(9x\\right)\\right)$, $16 D\\left(e^{-8x}\\cos\\!\\left(9x\\right)\\right)$, and $145 e^{-8x}\\cos\\!\\left(9x\\right)$. Place the coefficients from the terms $e^{-8x}\\cos(9x)$ and $e^{-8x}\\sin(9x)$ in the table. The columns of the table should add to zero.\n$\\begin{array}{ccc}\\hline & e^{-8x}\\cos\\!\\left(9x\\right) & e^{-8x}\\sin\\!\\left(9x\\right) \\\\ \\hline 145 & [ANS] & [ANS] \\\\ \\hline 16 D & [ANS] & [ANS] \\\\ \\hline D^2 & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "-8", "9", "145", "0", "-128", "-144", "-17", "144" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "The differential operator $D^2+8 D+52$ has the form $D^2-2\\alpha D+\\alpha^2+\\beta^2$ where $\\alpha=$ [ANS] and $\\beta=$ [ANS].\nTherefore $D^2+8 D+52$ should annihilate the function $f=e^{-4x}\\cos\\!\\left(6x\\right)$ and $g=e^{-4x}\\sin\\!\\left(6x\\right)$. We will check that for $e^{-4x}\\cos\\!\\left(6x\\right)$. Note that when we compute the derivative and second derivative we will get terms that have $e^{-4x}\\sin\\!\\left(6x\\right)$ in them, so we will have to account for them below.\nCompute $D^2\\left(e^{-4x}\\cos\\!\\left(6x\\right)\\right)$, $8 D\\left(e^{-4x}\\cos\\!\\left(6x\\right)\\right)$, and $52 e^{-4x}\\cos\\!\\left(6x\\right)$. Place the coefficients from the terms $e^{-4x}\\cos(6x)$ and $e^{-4x}\\sin(6x)$ in the table. The columns of the table should add to zero.\n$\\begin{array}{ccc}\\hline & e^{-4x}\\cos\\!\\left(6x\\right) & e^{-4x}\\sin\\!\\left(6x\\right) \\\\ \\hline 52 & [ANS] & [ANS] \\\\ \\hline 8 D & [ANS] & [ANS] \\\\ \\hline D^2 & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "-4", "6", "52", "0", "-32", "-48", "-20", "48" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Differential_equations_0224", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Differential operators", "level": "2", "keywords": [ "differential", "equation", "annihilator' 'constant' 'coefficient" ], "problem_v1": "The differential operator $(D^2-10 D+61)^3$ annihilates the functions [ANS].", "answer_v1": [ "(x^0*e^(5*x)*cos(6*x), x^0*e^(5*x)*sin(6*x), x^1*e^(5*x)*cos(6*x), x^1*e^(5*x)*sin(6*x), x^2*e^(5*x)*cos(6*x), x^2*e^(5*x)*sin(6*x))" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "The differential operator $(D^2+16 D+145)^3$ annihilates the functions [ANS].", "answer_v2": [ "(x^0*e^(-8*x)*cos(9*x), x^0*e^(-8*x)*sin(9*x), x^1*e^(-8*x)*cos(9*x), x^1*e^(-8*x)*sin(9*x), x^2*e^(-8*x)*cos(9*x), x^2*e^(-8*x)*sin(9*x))" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "The differential operator $(D^2+8 D+52)^3$ annihilates the functions [ANS].", "answer_v3": [ "(x^0*e^(-4*x)*cos(6*x), x^0*e^(-4*x)*sin(6*x), x^1*e^(-4*x)*cos(6*x), x^1*e^(-4*x)*sin(6*x), x^2*e^(-4*x)*cos(6*x), x^2*e^(-4*x)*sin(6*x))" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0225", "subject": "Differential_equations", "topic": "Higher order differential equations", "subtopic": "Differential operators", "level": "3", "keywords": [ "differential", "equation", "annihilator' 'constant' 'coefficient" ], "problem_v1": "The differential operator $(D-5)^3$ is supposed to annihilate the function $x^{2}e^{5x}$, we will check that: $(D)(x^{2}e^{5x})=$ [ANS]\n$(D-5)(x^{2}e^{5x})=$ [ANS]\n$D((D-5)(x^{2}e^{5x}))=$ [ANS]\n$(D-5)^2(x^{2}e^{5x})=$ [ANS]\n$D((D-5)^2(x^{2}e^{5x}))=$ [ANS]\n$(D-5)^3(x^{2}e^{5x})=$ [ANS]", "answer_v1": [ "2*x*e^(5*x)+x^2*5*e^(5*x)*ln(e)", "2*x*e^(5*x)", "2*e^(5*x)+2*x*5*e^(5*x)*ln(e)", "2*e^(5*x)", "2*5*e^(5*x)*ln(e)", "0" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "The differential operator $(D+8)^3$ is supposed to annihilate the function $x^{2}e^{-8x}$, we will check that: $(D)(x^{2}e^{-8x})=$ [ANS]\n$(D+8)(x^{2}e^{-8x})=$ [ANS]\n$D((D+8)(x^{2}e^{-8x}))=$ [ANS]\n$(D+8)^2(x^{2}e^{-8x})=$ [ANS]\n$D((D+8)^2(x^{2}e^{-8x}))=$ [ANS]\n$(D+8)^3(x^{2}e^{-8x})=$ [ANS]", "answer_v2": [ "2*x*e^(-8*x)-x^2*8*e^(-8*x)*ln(e)", "2*x*e^(-8*x)", "2*e^(-8*x)-2*x*8*e^(-8*x)*ln(e)", "2*e^(-8*x)", "-2*8*e^(-8*x)*ln(e)", "0" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "The differential operator $(D+4)^3$ is supposed to annihilate the function $x^{2}e^{-4x}$, we will check that: $(D)(x^{2}e^{-4x})=$ [ANS]\n$(D+4)(x^{2}e^{-4x})=$ [ANS]\n$D((D+4)(x^{2}e^{-4x}))=$ [ANS]\n$(D+4)^2(x^{2}e^{-4x})=$ [ANS]\n$D((D+4)^2(x^{2}e^{-4x}))=$ [ANS]\n$(D+4)^3(x^{2}e^{-4x})=$ [ANS]", "answer_v3": [ "2*x*e^(-4*x)-x^2*4*e^(-4*x)*ln(e)", "2*x*e^(-4*x)", "2*e^(-4*x)-2*x*4*e^(-4*x)*ln(e)", "2*e^(-4*x)", "-2*4*e^(-4*x)*ln(e)", "0" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Differential_equations_0226", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Applications and solving differential equations", "level": "3", "keywords": [ "laplace", "differential", "equation" ], "problem_v1": "Take the Laplace transform of the IVP $5 \\frac{dy}{dt} -y=0, y(0)=2$ Use $Y$ for the Laplace transform of $y$, (not $Y(s)$). [ANS]=[ANS]\nSo $\\begin{array}{ccc}\\hline Y=& & \\frac{[ANS]}{s-[ANS]} \\\\ \\hline \\end{array}$\nand $y(t)=$ [ANS]", "answer_v1": [ "(-10)+(5*s-1)*Y", "0", "2", "0.2", "2*e^(t/5)" ], "answer_type_v1": [ "EX", "NV", "NV", "NV", "EX" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Take the Laplace transform of the IVP $-9 \\frac{dy}{dt} -y=0, y(0)=9$ Use $Y$ for the Laplace transform of $y$, (not $Y(s)$). [ANS]=[ANS]\nSo $\\begin{array}{ccc}\\hline Y=& & \\frac{[ANS]}{s-[ANS]} \\\\ \\hline \\end{array}$\nand $y(t)=$ [ANS]", "answer_v2": [ "81+(-9*s-1)*Y", "0", "9", "-0.111111", "9*e^[t/(-9)]" ], "answer_type_v2": [ "EX", "NV", "NV", "NV", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Take the Laplace transform of the IVP $-4 \\frac{dy}{dt} -y=0, y(0)=2$ Use $Y$ for the Laplace transform of $y$, (not $Y(s)$). [ANS]=[ANS]\nSo $\\begin{array}{ccc}\\hline Y=& & \\frac{[ANS]}{s-[ANS]} \\\\ \\hline \\end{array}$\nand $y(t)=$ [ANS]", "answer_v3": [ "8+(-4*s-1)*Y", "0", "2", "-0.25", "2*e^[t/(-4)]" ], "answer_type_v3": [ "EX", "NV", "NV", "NV", "EX" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Differential_equations_0227", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Applications and solving differential equations", "level": "3", "keywords": [ "Laplace", "transform" ], "problem_v1": "Transform the differential equation y''+y'+2y=\\cos\\!\\left(at\\right) y(0)=-4 y^\\prime=-3 into an algebraic equation by taking the Laplace transform of each side. [ANS] $=$ [ANS]\nTherefore $Y=$ [ANS]", "answer_v1": [ "-1*s*(-4)-(-3)-(-4)+(1*s^2+1*s+2)*Y", "s/(a^2+s^2)", "[s-(a^2+s^2)*(4*s+3+4)]/[(s^2+s+2)*(a^2+s^2)]" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Transform the differential equation 5y''-4y'-2y=t^{1} y(0)=8 y^\\prime=-3 into an algebraic equation by taking the Laplace transform of each side. [ANS] $=$ [ANS]\nTherefore $Y=$ [ANS]", "answer_v2": [ "-5*s*8-(-15)-(-32)+[5*s^2+(-4)*s+(-2)]*Y", "1/(s^2)", "[1+s^2*(8*5*s-15-32)]/[(5*s^2-4*s-2)*s^2]" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Transform the differential equation y''-2y'+y=\\sinh\\!\\left(at\\right) y(0)=-6 y^\\prime=-3 into an algebraic equation by taking the Laplace transform of each side. [ANS] $=$ [ANS]\nTherefore $Y=$ [ANS]", "answer_v3": [ "-1*s*(-6)-(-3)-12+[1*s^2+(-2)*s+1]*Y", "-a/(a^2-s^2)", "[(a^2-s^2)*[12-(6*s+3)]-a]/[(s^2-2*s+1)*(a^2-s^2)]" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0228", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Applications and solving differential equations", "level": "3", "keywords": [ "integrals", "integration by parts" ], "problem_v1": "Solve the IVP \\frac{d^2y}{dt^2} +36 y=\\delta(t-k\\pi), y(0)=0,y^\\prime(0)=2. The Laplace transform of the solutions is L{y}=[ANS]\nThe general solution is y=[ANS]", "answer_v1": [ "[e^{-k*pi*s}+2]/(s^2+36)", "[u(t-k*pi)*sin(6*(t-k*pi))+2*sin(6*t)]/6" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Solve the IVP \\frac{d^2y}{dt^2} +4 y=\\delta(t-k\\pi), y(0)=0,y^\\prime(0)=8. The Laplace transform of the solutions is L{y}=[ANS]\nThe general solution is y=[ANS]", "answer_v2": [ "[e^{-k*pi*s}+8]/(s^2+4)", "[u(t-k*pi)*sin(2*(t-k*pi))+8*sin(2*t)]/2" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Solve the IVP \\frac{d^2y}{dt^2} +9 y=\\delta(t-k\\pi), y(0)=0,y^\\prime(0)=2. The Laplace transform of the solutions is L{y}=[ANS]\nThe general solution is y=[ANS]", "answer_v3": [ "[e^{-k*pi*s}+2]/(s^2+9)", "[u(t-k*pi)*sin(3*(t-k*pi))+2*sin(3*t)]/3" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0229", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Applications and solving differential equations", "level": "5", "keywords": [ "differential equations", "first order", "integrating factor" ], "problem_v1": "Assume that two colonies each have $1300$ members at time $t=0$ and that each evolves with a constant relative birth rate $k=r_{b}-r_{d}$. For colony 1, assume that individuals migrate into the colony at a rate of $70$ individuals per unit time. Assume that this immigration occurs for $0 \\leq t \\leq 1$ and ceases thereafter. For colony 2, assume that a similar migration pattern occurs but is delayed by one unit of time; that is, individuals immigrate at a rate of $70$ individuals per unit time, $1 \\leq t \\leq 2$. Suppose we are interested in comparing the evolution of these two populations over the time interval $0 \\leq t \\leq 2$. The initial value problems governing the two populations are\n\\begin{array}{lllll} \\frac{dP_1}{dt} =k P_1+M_1(t), && P_1(0)=1300, && M_1(t)=\\left\\lbrace \\begin{array}{rcl} 70, && 0 \\leq t \\leq 1, \\\\ 0, && 1 < t \\leq 2. \\\\ \\end{array} \\right. \\\\ \\\\ \\frac{dP_2}{dt} =k P_2+M_2(t), && P_2(0)=1300, && M_2(t)=\\left\\lbrace \\begin{array}{rcl} 0, && 0 \\leq t < 1, \\\\ 70, && 1 \\leq t \\leq 2. \\\\ \\end{array} \\right. \\end{array}\nSolve both problems to find $P_1$ and $P_2$ at time $t=2$. $P_1(2)=$ [ANS]\n$P_2(2)=$ [ANS]\nShow that $ P_1(2)-P_2(2)= \\frac{70}{k} \\left(e^k-1 \\right)^2$. If $k > 0$, which population is larger at time $t=2$? [ANS] If $k < 0$, which population is larger at time $t=2$? [ANS]\nSuppose that there is a fixed number of individuals that can be introduced into a population at any time through migration and that the objective is to maximize the population at some fixed future time. Do the calculations performed in this problem suggest a strategy (based on the relative birth rate) for accomplishing this? Suppose that there is a fixed number of individuals that can be introduced into a population at any time through migration and that the objective is to maximize the population at some fixed future time. Do the calculations performed in this problem suggest a strategy (based on the relative birth rate) for accomplishing this?", "answer_v1": [ "e^k*[(1300*k+70)*e^k-70]/k", "([1300*k+70*e^{-k}]*e^{2*k}-70)/k", "colony 1", "colony 2" ], "answer_type_v1": [ "EX", "EX", "MCS", "MCS" ], "options_v1": [ [], [], [ "colony 1", "colony 2", "cannot be determined" ], [ "colony 1", "colony 2", "cannot be determined" ] ], "problem_v2": "Assume that two colonies each have $500$ members at time $t=0$ and that each evolves with a constant relative birth rate $k=r_{b}-r_{d}$. For colony 1, assume that individuals migrate into the colony at a rate of $100$ individuals per unit time. Assume that this immigration occurs for $0 \\leq t \\leq 1$ and ceases thereafter. For colony 2, assume that a similar migration pattern occurs but is delayed by one unit of time; that is, individuals immigrate at a rate of $100$ individuals per unit time, $1 \\leq t \\leq 2$. Suppose we are interested in comparing the evolution of these two populations over the time interval $0 \\leq t \\leq 2$. The initial value problems governing the two populations are\n\\begin{array}{lllll} \\frac{dP_1}{dt} =k P_1+M_1(t), && P_1(0)=500, && M_1(t)=\\left\\lbrace \\begin{array}{rcl} 100, && 0 \\leq t \\leq 1, \\\\ 0, && 1 < t \\leq 2. \\\\ \\end{array} \\right. \\\\ \\\\ \\frac{dP_2}{dt} =k P_2+M_2(t), && P_2(0)=500, && M_2(t)=\\left\\lbrace \\begin{array}{rcl} 0, && 0 \\leq t < 1, \\\\ 100, && 1 \\leq t \\leq 2. \\\\ \\end{array} \\right. \\end{array}\nSolve both problems to find $P_1$ and $P_2$ at time $t=2$. $P_1(2)=$ [ANS]\n$P_2(2)=$ [ANS]\nShow that $ P_1(2)-P_2(2)= \\frac{100}{k} \\left(e^k-1 \\right)^2$. If $k > 0$, which population is larger at time $t=2$? [ANS] If $k < 0$, which population is larger at time $t=2$? [ANS]\nSuppose that there is a fixed number of individuals that can be introduced into a population at any time through migration and that the objective is to maximize the population at some fixed future time. Do the calculations performed in this problem suggest a strategy (based on the relative birth rate) for accomplishing this? Suppose that there is a fixed number of individuals that can be introduced into a population at any time through migration and that the objective is to maximize the population at some fixed future time. Do the calculations performed in this problem suggest a strategy (based on the relative birth rate) for accomplishing this?", "answer_v2": [ "e^k*[(500*k+100)*e^k-100]/k", "([500*k+100*e^{-k}]*e^{2*k}-100)/k", "colony 1", "colony 2" ], "answer_type_v2": [ "EX", "EX", "MCS", "MCS" ], "options_v2": [ [], [], [ "colony 1", "colony 2", "cannot be determined" ], [ "colony 1", "colony 2", "cannot be determined" ] ], "problem_v3": "Assume that two colonies each have $800$ members at time $t=0$ and that each evolves with a constant relative birth rate $k=r_{b}-r_{d}$. For colony 1, assume that individuals migrate into the colony at a rate of $70$ individuals per unit time. Assume that this immigration occurs for $0 \\leq t \\leq 1$ and ceases thereafter. For colony 2, assume that a similar migration pattern occurs but is delayed by one unit of time; that is, individuals immigrate at a rate of $70$ individuals per unit time, $1 \\leq t \\leq 2$. Suppose we are interested in comparing the evolution of these two populations over the time interval $0 \\leq t \\leq 2$. The initial value problems governing the two populations are\n\\begin{array}{lllll} \\frac{dP_1}{dt} =k P_1+M_1(t), && P_1(0)=800, && M_1(t)=\\left\\lbrace \\begin{array}{rcl} 70, && 0 \\leq t \\leq 1, \\\\ 0, && 1 < t \\leq 2. \\\\ \\end{array} \\right. \\\\ \\\\ \\frac{dP_2}{dt} =k P_2+M_2(t), && P_2(0)=800, && M_2(t)=\\left\\lbrace \\begin{array}{rcl} 0, && 0 \\leq t < 1, \\\\ 70, && 1 \\leq t \\leq 2. \\\\ \\end{array} \\right. \\end{array}\nSolve both problems to find $P_1$ and $P_2$ at time $t=2$. $P_1(2)=$ [ANS]\n$P_2(2)=$ [ANS]\nShow that $ P_1(2)-P_2(2)= \\frac{70}{k} \\left(e^k-1 \\right)^2$. If $k > 0$, which population is larger at time $t=2$? [ANS] If $k < 0$, which population is larger at time $t=2$? [ANS]\nSuppose that there is a fixed number of individuals that can be introduced into a population at any time through migration and that the objective is to maximize the population at some fixed future time. Do the calculations performed in this problem suggest a strategy (based on the relative birth rate) for accomplishing this? Suppose that there is a fixed number of individuals that can be introduced into a population at any time through migration and that the objective is to maximize the population at some fixed future time. Do the calculations performed in this problem suggest a strategy (based on the relative birth rate) for accomplishing this?", "answer_v3": [ "e^k*[(800*k+70)*e^k-70]/k", "([800*k+70*e^{-k}]*e^{2*k}-70)/k", "colony 1", "colony 2" ], "answer_type_v3": [ "EX", "EX", "MCS", "MCS" ], "options_v3": [ [], [], [ "colony 1", "colony 2", "cannot be determined" ], [ "colony 1", "colony 2", "cannot be determined" ] ] }, { "id": "Differential_equations_0230", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Applications and solving differential equations", "level": "5", "keywords": [ "Laplace transform" ], "problem_v1": "A lake containing 140 million gallons of fresh water has a stream flowing through it. Water enters the lake at a constant rate of 7 million gal/day and leaves at the same rate. An upstream manufacturer begins to discharge pollutants into the feeder stream. Each day, during the hours from 8 a.m. to 8 p.m., the stream has a pollutant concentration of 3 mg/gal $(3 \\times 10^{-6}$ kg/gal $)$ ; at other times, the stream feeds in fresh water. Assume that a well-stirred mixture leaves the lake and that the manufacturer operates seven days per week.\nLet $t=0$ denote the instant that pollutants first enter the lake. Let $q(t)$ denote the amount of pollutant (in kilograms) present in the lake at time $t$ (in days). Use a \"conservation of pollutant\" principle (rate of change=rate in-rate out) to formulate the initial value problem satisfied by $q(t)$:\n$q'=$ [ANS] $\\cdot c_i \\-\\ $ [ANS] $\\cdot q, \\quad \\quad q(0)=$ [ANS]\nwhere $\\begin{array}{ccc}\\hline c_i= \\Bigg\\lbrace & & [ANS]if 0 \\leq t < \\frac{1}{2} , [ANS]if \\frac{1}{2} \\leq t < 1. \\\\ \\hline \\end{array}$\nand $c_i(t+$ [ANS] $)=c_i(t)$.\nTake the Laplace transform of both sides of the differential equation formulated in part (a) to determine $Q(s)={\\mathcal L}\\left\\lbrace q(t) \\right\\rbrace$.\n$ Q(s)={\\mathcal L}\\left\\lbrace q(t) \\right\\rbrace=$ [ANS]", "answer_v1": [ "7", "1/20", "0", "3", "0", "1", "7*3*[1-e^(-s/2)]/(s*[1-e^(-s)]*(s+1/20))" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "EX" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "A lake containing 80 million gallons of fresh water has a stream flowing through it. Water enters the lake at a constant rate of 4 million gal/day and leaves at the same rate. An upstream manufacturer begins to discharge pollutants into the feeder stream. Each day, during the hours from 6 a.m. to 6 p.m., the stream has a pollutant concentration of 2 mg/gal $(2 \\times 10^{-6}$ kg/gal $)$ ; at other times, the stream feeds in fresh water. Assume that a well-stirred mixture leaves the lake and that the manufacturer operates seven days per week.\nLet $t=0$ denote the instant that pollutants first enter the lake. Let $q(t)$ denote the amount of pollutant (in kilograms) present in the lake at time $t$ (in days). Use a \"conservation of pollutant\" principle (rate of change=rate in-rate out) to formulate the initial value problem satisfied by $q(t)$:\n$q'=$ [ANS] $\\cdot c_i \\-\\ $ [ANS] $\\cdot q, \\quad \\quad q(0)=$ [ANS]\nwhere $\\begin{array}{ccc}\\hline c_i= \\Bigg\\lbrace & & [ANS]if 0 \\leq t < \\frac{1}{2} , [ANS]if \\frac{1}{2} \\leq t < 1. \\\\ \\hline \\end{array}$\nand $c_i(t+$ [ANS] $)=c_i(t)$.\nTake the Laplace transform of both sides of the differential equation formulated in part (a) to determine $Q(s)={\\mathcal L}\\left\\lbrace q(t) \\right\\rbrace$.\n$ Q(s)={\\mathcal L}\\left\\lbrace q(t) \\right\\rbrace=$ [ANS]", "answer_v2": [ "4", "1/20", "0", "2", "0", "1", "4*2*[1-e^(-s/2)]/(s*[1-e^(-s)]*(s+1/20))" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "EX" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "A lake containing 100 million gallons of fresh water has a stream flowing through it. Water enters the lake at a constant rate of 5 million gal/day and leaves at the same rate. An upstream manufacturer begins to discharge pollutants into the feeder stream. Each day, during the hours from 7 a.m. to 7 p.m., the stream has a pollutant concentration of 2 mg/gal $(2 \\times 10^{-6}$ kg/gal $)$ ; at other times, the stream feeds in fresh water. Assume that a well-stirred mixture leaves the lake and that the manufacturer operates seven days per week.\nLet $t=0$ denote the instant that pollutants first enter the lake. Let $q(t)$ denote the amount of pollutant (in kilograms) present in the lake at time $t$ (in days). Use a \"conservation of pollutant\" principle (rate of change=rate in-rate out) to formulate the initial value problem satisfied by $q(t)$:\n$q'=$ [ANS] $\\cdot c_i \\-\\ $ [ANS] $\\cdot q, \\quad \\quad q(0)=$ [ANS]\nwhere $\\begin{array}{ccc}\\hline c_i= \\Bigg\\lbrace & & [ANS]if 0 \\leq t < \\frac{1}{2} , [ANS]if \\frac{1}{2} \\leq t < 1. \\\\ \\hline \\end{array}$\nand $c_i(t+$ [ANS] $)=c_i(t)$.\nTake the Laplace transform of both sides of the differential equation formulated in part (a) to determine $Q(s)={\\mathcal L}\\left\\lbrace q(t) \\right\\rbrace$.\n$ Q(s)={\\mathcal L}\\left\\lbrace q(t) \\right\\rbrace=$ [ANS]", "answer_v3": [ "5", "1/20", "0", "2", "0", "1", "5*2*[1-e^(-s/2)]/(s*[1-e^(-s)]*(s+1/20))" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "EX" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Differential_equations_0231", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Applications and solving differential equations", "level": "5", "keywords": [ "Laplace transform" ], "problem_v1": "Consider the initial value problem for $0 < t < \\infty$: ay''+by'+cy=f(t), \\quad \\quad y(0)=0, \\quad y'(0)=0, where $a, b, c$ are constants and $f(t)$ is a known function. We can view this problem as defining a linear system, where $f(t)$ is a known input and the corresponding solution $y(t)$ is the output. Laplace transforms of the input and output functions satisfy the multiplicative relation Y(s)=\\Phi(s) F(s), where $\\Phi(s)= \\frac{1}{as^2+bs+c} $ is the system transfer function.\nSuppose an input $f(t)=6t$, when applied to the linear system above, produces the output $y(t)=5\\!\\left(e^{-3t}-1\\right)+t\\!\\left(e^{-3t}+14\\right), \\ t\\geq 0.$\nFind $Y(s)={\\mathcal L}\\left\\lbrace y(t) \\right\\rbrace$ and $F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace$.\n$ Y(s)=$ [ANS]\n$ F(s)=$ [ANS]\nUse your answer to part (a) to find the system transfer function, $\\Phi(s)$.\n$ \\Phi(s)=$ [ANS]\nWhat will be the output if a Heaviside unit step input $f(t)=h(t)$ is applied to the system?\nNew $ y(t)=$ [ANS]", "answer_v1": [ "5*[1/(s+3)-1/s]+1/[(s+3)^2]+14/(s^2)", "6/(s^2)", "(5*[1/(s+3)-1/s]+1/[(s+3)^2]+14/(s^2))/[6/(s^2)]", "[14*h(t)-[14*e^{-3*t}+3*e^{-3*t}*t]]/6" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the initial value problem for $0 < t < \\infty$: ay''+by'+cy=f(t), \\quad \\quad y(0)=0, \\quad y'(0)=0, where $a, b, c$ are constants and $f(t)$ is a known function. We can view this problem as defining a linear system, where $f(t)$ is a known input and the corresponding solution $y(t)$ is the output. Laplace transforms of the input and output functions satisfy the multiplicative relation Y(s)=\\Phi(s) F(s), where $\\Phi(s)= \\frac{1}{as^2+bs+c} $ is the system transfer function.\nSuppose an input $f(t)=4t$, when applied to the linear system above, produces the output $y(t)=2\\!\\left(e^{-4t}-1\\right)+t\\!\\left(e^{-4t}+7\\right), \\ t\\geq 0.$\nFind $Y(s)={\\mathcal L}\\left\\lbrace y(t) \\right\\rbrace$ and $F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace$.\n$ Y(s)=$ [ANS]\n$ F(s)=$ [ANS]\nUse your answer to part (a) to find the system transfer function, $\\Phi(s)$.\n$ \\Phi(s)=$ [ANS]\nWhat will be the output if a Heaviside unit step input $f(t)=h(t)$ is applied to the system?\nNew $ y(t)=$ [ANS]", "answer_v2": [ "2*[1/(s+4)-1/s]+1/[(s+4)^2]+7/(s^2)", "4/(s^2)", "(2*[1/(s+4)-1/s]+1/[(s+4)^2]+7/(s^2))/[4/(s^2)]", "[7*h(t)-[7*e^{-4*t}+4*e^{-4*t}*t]]/4" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the initial value problem for $0 < t < \\infty$: ay''+by'+cy=f(t), \\quad \\quad y(0)=0, \\quad y'(0)=0, where $a, b, c$ are constants and $f(t)$ is a known function. We can view this problem as defining a linear system, where $f(t)$ is a known input and the corresponding solution $y(t)$ is the output. Laplace transforms of the input and output functions satisfy the multiplicative relation Y(s)=\\Phi(s) F(s), where $\\Phi(s)= \\frac{1}{as^2+bs+c} $ is the system transfer function.\nSuppose an input $f(t)=5t$, when applied to the linear system above, produces the output $y(t)=3\\!\\left(e^{-2t}-1\\right)+t\\!\\left(e^{-2t}+5\\right), \\ t\\geq 0.$\nFind $Y(s)={\\mathcal L}\\left\\lbrace y(t) \\right\\rbrace$ and $F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace$.\n$ Y(s)=$ [ANS]\n$ F(s)=$ [ANS]\nUse your answer to part (a) to find the system transfer function, $\\Phi(s)$.\n$ \\Phi(s)=$ [ANS]\nWhat will be the output if a Heaviside unit step input $f(t)=h(t)$ is applied to the system?\nNew $ y(t)=$ [ANS]", "answer_v3": [ "3*[1/(s+2)-1/s]+1/[(s+2)^2]+5/(s^2)", "5/(s^2)", "(3*[1/(s+2)-1/s]+1/[(s+2)^2]+5/(s^2))/[5/(s^2)]", "[5*h(t)-[5*e^{-2*t}+2*e^{-2*t}*t]]/5" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0232", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Applications and solving differential equations", "level": "2", "keywords": [ "Laplace transform" ], "problem_v1": "Consider the initial value problem y'+4 y=48 t, \\quad \\quad y(0)=6. Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of $y(t)$ by $Y(s)$. Do not move any terms from one side of the equation to the other (until you get to part (b) below). [ANS] $=$ [ANS]\nSolve your equation for $Y(s)$.\n$ Y(s)={\\mathcal L}\\left\\lbrace y(t) \\right\\rbrace=$ [ANS]\nTake the inverse Laplace transform of both sides of the previous equation to solve for $y(t)$.\n$y(t)=$ [ANS]", "answer_v1": [ "s*Y(s)-6+4*Y(s)", "48/(s^2)", "48/[s^2*(s+4)]+6/(s+4)", "12*t-3+9*e^{-4*t}" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the initial value problem y'+2 y=20 t, \\quad \\quad y(0)=3. Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of $y(t)$ by $Y(s)$. Do not move any terms from one side of the equation to the other (until you get to part (b) below). [ANS] $=$ [ANS]\nSolve your equation for $Y(s)$.\n$ Y(s)={\\mathcal L}\\left\\lbrace y(t) \\right\\rbrace=$ [ANS]\nTake the inverse Laplace transform of both sides of the previous equation to solve for $y(t)$.\n$y(t)=$ [ANS]", "answer_v2": [ "s*Y(s)-3+2*Y(s)", "20/(s^2)", "20/[s^2*(s+2)]+3/(s+2)", "10*t-5+8*e^(-2*t)" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the initial value problem y'+2 y=16 t, \\quad \\quad y(0)=4. Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of $y(t)$ by $Y(s)$. Do not move any terms from one side of the equation to the other (until you get to part (b) below). [ANS] $=$ [ANS]\nSolve your equation for $Y(s)$.\n$ Y(s)={\\mathcal L}\\left\\lbrace y(t) \\right\\rbrace=$ [ANS]\nTake the inverse Laplace transform of both sides of the previous equation to solve for $y(t)$.\n$y(t)=$ [ANS]", "answer_v3": [ "s*Y(s)-4+2*Y(s)", "16/(s^2)", "16/[s^2*(s+2)]+4/(s+2)", "8*t-4+8*e^(-2*t)" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0233", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Applications and solving differential equations", "level": "5", "keywords": [], "problem_v1": "Take the Laplace transform of the following initial value problem and solve for $Y(s)=\\mathcal{L}\\lbrace y(t) \\rbrace$: y''+{10} y'+{41} y=\\begin{cases} t, & 0 \\leq t < 1 \\cr 0, & 1 \\leq t \\end{cases} \\hspace{0.5in} y(0)=0, \\; y'(0)=0 $Y(s)=$ [ANS]. Now find the inverse transform to find $y(t)=$ [ANS]\n$-u_1(t)$ times a similar function to the above and another one given by the second equation below. It's too long to enter in to a blank space. Note: \\frac{1}{s^2(s^2+{10} s+41)}= \\frac{-\\frac{10}{1681} }{s}+ \\frac{\\frac{1}{41} }{s^2}+ \\frac{\\frac{10}{1681} (s+5)+ \\frac{9}{1681} }{(s+5)^2+4^2} and \\frac{s}{s^2(s^2+{10} s+41)}= \\frac{\\frac{1}{41} }{s}+ \\frac{-\\frac{1}{41} (s+5)- \\frac{5}{41} }{(s+5)^2+4^2}", "answer_v1": [ "(1-exp(-s)*(1+s))/(s^2*(s^2+10*s+41))", "-0.00594883997620464+0.024390243902439*t+0.00594883997620464*exp(-5*t)*cos(4*t)+0.00133848899464604*exp(-5*t)*sin(4*t)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Take the Laplace transform of the following initial value problem and solve for $Y(s)=\\mathcal{L}\\lbrace y(t) \\rbrace$: y''+{4} y'+{29} y=\\begin{cases} t, & 0 \\leq t < 1 \\cr 0, & 1 \\leq t \\end{cases} \\hspace{0.5in} y(0)=0, \\; y'(0)=0 $Y(s)=$ [ANS]. Now find the inverse transform to find $y(t)=$ [ANS]\n$-u_1(t)$ times a similar function to the above and another one given by the second equation below. It's too long to enter in to a blank space. Note: \\frac{1}{s^2(s^2+{4} s+29)}= \\frac{-\\frac{4}{841} }{s}+ \\frac{\\frac{1}{29} }{s^2}+ \\frac{\\frac{4}{841} (s+2)+ \\frac{-21}{841} }{(s+2)^2+5^2} and \\frac{s}{s^2(s^2+{4} s+29)}= \\frac{\\frac{1}{29} }{s}+ \\frac{-\\frac{1}{29} (s+2)- \\frac{2}{29} }{(s+2)^2+5^2}", "answer_v2": [ "(1-exp(-s)*(1+s))/(s^2*(s^2+4*s+29))", "-0.00475624256837099+0.0344827586206897*t+0.00475624256837099*exp(-2*t)*cos(5*t) - 0.00499405469678954*exp(-2*t)*sin(5*t)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Take the Laplace transform of the following initial value problem and solve for $Y(s)=\\mathcal{L}\\lbrace y(t) \\rbrace$: y''+{6} y'+{25} y=\\begin{cases} t, & 0 \\leq t < 1 \\cr 0, & 1 \\leq t \\end{cases} \\hspace{0.5in} y(0)=0, \\; y'(0)=0 $Y(s)=$ [ANS]. Now find the inverse transform to find $y(t)=$ [ANS]\n$-u_1(t)$ times a similar function to the above and another one given by the second equation below. It's too long to enter in to a blank space. Note: \\frac{1}{s^2(s^2+{6} s+25)}= \\frac{-\\frac{6}{625} }{s}+ \\frac{\\frac{1}{25} }{s^2}+ \\frac{\\frac{6}{625} (s+3)+ \\frac{-7}{625} }{(s+3)^2+4^2} and \\frac{s}{s^2(s^2+{6} s+25)}= \\frac{\\frac{1}{25} }{s}+ \\frac{-\\frac{1}{25} (s+3)- \\frac{3}{25} }{(s+3)^2+4^2}", "answer_v3": [ "(1-exp(-s)*(1+s))/(s^2*(s^2+6*s+25))", "-0.0096+0.04*t+0.0096*exp(-3*t)*cos(4*t) - 0.0028*exp(-3*t)*sin(4*t)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0234", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Applications and solving differential equations", "level": "4", "keywords": [], "problem_v1": "Use the Laplace transform to solve the following initial value problem: y''+{25} y=8 \\delta(t-6) \\hspace{0.5in} y(0)=0, \\; y'(0)=0 Use step(t-c) for $u_c(t)$. $y(t)=$ [ANS].", "answer_v1": [ "step(t-6)*1.6*sin(5*(t-6))" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use the Laplace transform to solve the following initial value problem: y''+{4} y=6 \\delta(t-9) \\hspace{0.5in} y(0)=0, \\; y'(0)=0 Use step(t-c) for $u_c(t)$. $y(t)=$ [ANS].", "answer_v2": [ "step(t-9)*3*sin(2*(t-9))" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use the Laplace transform to solve the following initial value problem: y''+{9} y=7 \\delta(t-6) \\hspace{0.5in} y(0)=0, \\; y'(0)=0 Use step(t-c) for $u_c(t)$. $y(t)=$ [ANS].", "answer_v3": [ "step(t-6)*2.33333333333333*sin(3*(t-6))" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0235", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Basic transformations", "level": "2", "keywords": [ "integrals", "integration by parts" ], "problem_v1": "Find the Laplace transform of $4\\sin\\!\\left(at\\right)-4\\cos\\!\\left(at\\right)$: [ANS]", "answer_v1": [ "[4*a*(a^2+s^2)-4*s*(a^2+s^2)]/[(a^2+s^2)*(a^2+s^2)]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the Laplace transform of $8-3\\cosh\\!\\left(at\\right)$: [ANS]", "answer_v2": [ "[8*(s^2-a^2)-3*s*s]/[(s^2-a^2)*s]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the Laplace transform of $\\sin\\!\\left(at\\right)-6t^{3}$: [ANS]", "answer_v3": [ "[a*s^4-36*(a^2+s^2)]/[(a^2+s^2)*s^4]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0236", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Basic transformations", "level": "3", "keywords": [ "integrals", "integration by parts" ], "problem_v1": "To find the Laplace transform of $e^{2t}\\sin\\!\\left(at\\right)$ you would substitute [ANS] into $ \\frac{a}{a^{2} +s^{2}}$ which gives [ANS]", "answer_v1": [ "s-2", "a/[a^2+(s-2)^2]" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "To find the Laplace transform of $e^{-7t}\\cosh\\!\\left(at\\right)$ you would substitute [ANS] into $ \\frac{s}{s^{2} -a^{2}}$ which gives [ANS]", "answer_v2": [ "s-(-7)", "[s-(-7)]/([s-(-7)]^2-a^2)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "To find the Laplace transform of $e^{-4t}\\sinh\\!\\left(at\\right)$ you would substitute [ANS] into $ \\frac{a}{s^{2} -a^{2}}$ which gives [ANS]", "answer_v3": [ "s-(-4)", "a/([s-(-4)]^2-a^2)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0237", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Basic transformations", "level": "3", "keywords": [ "Laplace transform" ], "problem_v1": "Find the Laplace transform $F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace$ of the function $f(t)=\\left(6-t\\right)\\!\\left(h\\!\\left(t-4\\right)-h\\!\\left(t-7\\right)\\right)$, for $s \\neq 0$.\n$ F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace=$ [ANS]", "answer_v1": [ "2*e^(-4*s)/s+e^(-7*s)/s-e^(-4*s)/(s^2)+e^(-7*s)/(s^2)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the Laplace transform $F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace$ of the function $f(t)=\\left(2-t\\right)\\!\\left(h\\!\\left(t-1\\right)-h\\!\\left(t-5\\right)\\right)$, for $s \\neq 0$.\n$ F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace=$ [ANS]", "answer_v2": [ "e^(-s)/s+3*e^(-5*s)/s-e^(-s)/(s^2)+e^(-5*s)/(s^2)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the Laplace transform $F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace$ of the function $f(t)=\\left(3-t\\right)\\!\\left(h\\!\\left(t-2\\right)-h\\!\\left(t-5\\right)\\right)$, for $s \\neq 0$.\n$ F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace=$ [ANS]", "answer_v3": [ "e^(-2*s)/s+2*e^(-5*s)/s-e^(-2*s)/(s^2)+e^(-5*s)/(s^2)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0238", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Basic transformations", "level": "2", "keywords": [ "Laplace transform" ], "problem_v1": "Consider the rational function F(s)= \\frac{s^3-3}{(s^2+6)^2 (s+11)^2} . Select ALL terms below that occur in the general form of the complete partial fraction decomposition of $F(s)$. The capital letters A, B, C,..., L denote constants. [ANS] A. $ \\frac{Ks+L}{(s+11)^2} $ B. $ \\frac{G}{s+11} $ C. $ \\frac{J}{(s+11)^2} $ D. $ \\frac{A}{s^2+6} $ E. $ \\frac{D}{(s^2+6)^2} $ F. $ \\frac{Es+F}{(s^2+6)^2} $ G. $ \\frac{Bs+C}{s^2+6} $ H. $ \\frac{Hs+I}{s+11} $", "answer_v1": [ "BCFG" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v2": "Consider the rational function F(s)= \\frac{s^3-1}{(s^2+7)^2 (s+9)^2} . Select ALL terms below that occur in the general form of the complete partial fraction decomposition of $F(s)$. The capital letters A, B, C,..., L denote constants. [ANS] A. $ \\frac{Hs+I}{s+9} $ B. $ \\frac{J}{(s+9)^2} $ C. $ \\frac{A}{s^2+7} $ D. $ \\frac{Ks+L}{(s+9)^2} $ E. $ \\frac{Bs+C}{s^2+7} $ F. $ \\frac{D}{(s^2+7)^2} $ G. $ \\frac{Es+F}{(s^2+7)^2} $ H. $ \\frac{G}{s+9} $", "answer_v2": [ "BEGH" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v3": "Consider the rational function F(s)= \\frac{s^3-1}{(s^2+6)^2 (s+10)^2} . Select ALL terms below that occur in the general form of the complete partial fraction decomposition of $F(s)$. The capital letters A, B, C,..., L denote constants. [ANS] A. $ \\frac{Bs+C}{s^2+6} $ B. $ \\frac{A}{s^2+6} $ C. $ \\frac{G}{s+10} $ D. $ \\frac{J}{(s+10)^2} $ E. $ \\frac{D}{(s^2+6)^2} $ F. $ \\frac{Hs+I}{s+10} $ G. $ \\frac{Ks+L}{(s+10)^2} $ H. $ \\frac{Es+F}{(s^2+6)^2} $", "answer_v3": [ "ACDH" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G", "H" ] ] }, { "id": "Differential_equations_0239", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Basic transformations", "level": "4", "keywords": [], "problem_v1": "Given that \\mathcal{L} \\lbrace \\frac{\\cos(8 t)}{\\sqrt{\\pi t} } \\rbrace= \\frac{e^{-8/s}}{\\sqrt{s} } find the Laplace transform of $\\hspace{0.05in} \\sqrt{ \\frac{t}{\\pi} }\\cos(8 t)$. $\\mathcal{L} \\lbrace \\sqrt{ \\frac{t}{\\pi} }\\cos(8 t) \\rbrace=$ [ANS].", "answer_v1": [ "exp(-8/s)*(s-2*8)/(2*s^(5/2))" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Given that \\mathcal{L} \\lbrace \\frac{\\cos(2 t)}{\\sqrt{\\pi t} } \\rbrace= \\frac{e^{-2/s}}{\\sqrt{s} } find the Laplace transform of $\\hspace{0.05in} \\sqrt{ \\frac{t}{\\pi} }\\cos(2 t)$. $\\mathcal{L} \\lbrace \\sqrt{ \\frac{t}{\\pi} }\\cos(2 t) \\rbrace=$ [ANS].", "answer_v2": [ "exp(-2/s)*(s-2*2)/(2*s^(5/2))" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Given that \\mathcal{L} \\lbrace \\frac{\\cos(4 t)}{\\sqrt{\\pi t} } \\rbrace= \\frac{e^{-4/s}}{\\sqrt{s} } find the Laplace transform of $\\hspace{0.05in} \\sqrt{ \\frac{t}{\\pi} }\\cos(4 t)$. $\\mathcal{L} \\lbrace \\sqrt{ \\frac{t}{\\pi} }\\cos(4 t) \\rbrace=$ [ANS].", "answer_v3": [ "exp(-4/s)*(s-2*4)/(2*s^(5/2))" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0240", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Basic transformations", "level": "3", "keywords": [], "problem_v1": "Find the inverse Laplace transform of \\frac{8 s+6}{s^2+19} \\hspace{0.5in} s > 0 $y(t)=$ [ANS].", "answer_v1": [ "8*cos(4.35889894354067*t)+1.37649440322337*sin(4.35889894354067*t)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the inverse Laplace transform of \\frac{2 s+9}{s^2+13} \\hspace{0.5in} s > 0 $y(t)=$ [ANS].", "answer_v2": [ "2*cos(3.60555127546399*t)+2.49615088301353*sin(3.60555127546399*t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the inverse Laplace transform of \\frac{4 s+6}{s^2+13} \\hspace{0.5in} s > 0 $y(t)=$ [ANS].", "answer_v3": [ "4*cos(3.60555127546399*t)+1.66410058867569*sin(3.60555127546399*t)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0241", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Inverse transformations", "level": "3", "keywords": [ "integrals", "integration by parts" ], "problem_v1": "$\\begin{array}{ccc}\\hline \\frac{1}{s^{2} +4s+53}=& & \\frac{1}{\\big(s+[ANS] \\big)^2+[ANS] ^2} \\\\ \\hline \\end{array}$\n$ \\frac{1}{s^{2} +4s+53}=F\\Big\\vert_{s+2}$ where $F(s)=$ [ANS]\nTherefore $f(t)=$ [ANS]", "answer_v1": [ "2", "7", "1/(s^2+49)", "e^(-2*t)*sin(7*t)/7" ], "answer_type_v1": [ "NV", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "$\\begin{array}{ccc}\\hline \\frac{1}{s^{2} +16s+65}=& & \\frac{1}{\\big(s+[ANS] \\big)^2+[ANS] ^2} \\\\ \\hline \\end{array}$\n$ \\frac{1}{s^{2} +16s+65}=F\\Big\\vert_{s+8}$ where $F(s)=$ [ANS]\nTherefore $f(t)=$ [ANS]", "answer_v2": [ "8", "1", "1/(s^2+1)", "e^(-8*t)*sin(t)" ], "answer_type_v2": [ "NV", "NV", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "$\\begin{array}{ccc}\\hline \\frac{1}{s^{2} +4s+13}=& & \\frac{1}{\\big(s+[ANS] \\big)^2+[ANS] ^2} \\\\ \\hline \\end{array}$\n$ \\frac{1}{s^{2} +4s+13}=F\\Big\\vert_{s+2}$ where $F(s)=$ [ANS]\nTherefore $f(t)=$ [ANS]", "answer_v3": [ "2", "3", "1/(s^2+9)", "e^(-2*t)*sin(3*t)/3" ], "answer_type_v3": [ "NV", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0242", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Inverse transformations", "level": "3", "keywords": [ "laplace", "convolution" ], "problem_v1": "If $L\\lbrace f\\rbrace (s)=- \\frac{aa}{\\left(a^{2} +s^{2}\\right)\\!\\left(a^{2}-s^{2}\\right)}$ then\n$\\begin{array}{ccccccc}\\hline f(t)=& & [ANS] \\int [ANS] & & [ANS] & & d\\tau \\\\ \\hline \\end{array}$\nNote $\\tau$ is typed as tau.", "answer_v1": [ "0", "t", "sin(a*tau)*sinh(a*(t-tau))" ], "answer_type_v1": [ "NV", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "If $L\\lbrace f\\rbrace (s)= \\frac{s}{\\left(s^{2} -a^{2}\\right)s^{2}}$ then\n$\\begin{array}{ccccccc}\\hline f(t)=& & [ANS] \\int [ANS] & & [ANS] & & d\\tau \\\\ \\hline \\end{array}$\nNote $\\tau$ is typed as tau.", "answer_v2": [ "0", "t", "cosh(a*tau)*(t-tau)^1" ], "answer_type_v2": [ "NV", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "If $L\\lbrace f\\rbrace (s)= \\frac{a}{\\left(a^{2} +s^{2}\\right)\\!\\left(s-m\\right)}$ then\n$\\begin{array}{ccccccc}\\hline f(t)=& & [ANS] \\int [ANS] & & [ANS] & & d\\tau \\\\ \\hline \\end{array}$\nNote $\\tau$ is typed as tau.", "answer_v3": [ "0", "t", "sin(a*tau)*e^[m*(t-tau)]" ], "answer_type_v3": [ "NV", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0243", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Inverse transformations", "level": "3", "keywords": [ "integrals", "integration by parts" ], "problem_v1": "$ \\frac{a}{a^{2} +\\left(s+4\\right)^{2}}=F\\Big\\vert_{s+4}$ where $F(s)=$ [ANS]\nTherefore the inverse Laplace transform of $ \\frac{a}{a^{2} +\\left(s+4\\right)^{2}}$ is [ANS]", "answer_v1": [ "a/(a^2+s^2)", "e^{-4*t}*sin(a*t)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "$ \\frac{s-8}{\\left(s-8\\right)^{2} -a^{2}}=F\\Big\\vert_{s-8}$ where $F(s)=$ [ANS]\nTherefore the inverse Laplace transform of $ \\frac{s-8}{\\left(s-8\\right)^{2} -a^{2}}$ is [ANS]", "answer_v2": [ "s/(s^2-a^2)", "e^(8*t)*cosh(a*t)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "$- \\frac{a}{a^{2} -\\left(s+6\\right)^{2}}=F\\Big\\vert_{s+6}$ where $F(s)=$ [ANS]\nTherefore the inverse Laplace transform of $- \\frac{a}{a^{2} -\\left(s+6\\right)^{2}}$ is [ANS]", "answer_v3": [ "-[a/(a^2-s^2)]", "e^(-6*t)*sinh(a*t)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0244", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Inverse transformations", "level": "3", "keywords": [ "Laplace transform" ], "problem_v1": "Consider the function $ F(s)= \\frac{4s+3}{s^{3} -3s^{2}+3s-1}$.\nFind the partial fraction decomposition of $F(s)$:\n$ \\frac{4s+3}{s^{3} -3s^{2}+3s-1}=$ [ANS]+[ANS]\nFind the inverse Laplace transform of $ F(s)$.\n$ f(t)={\\mathcal L}^{-1} \\left\\lbrace F(s) \\right\\rbrace=$ [ANS]", "answer_v1": [ "7/[(s-1)^3]", "4/[(s-1)^2]", "7/2*e^t*t^2+4*e^t*t" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Consider the function $ F(s)= \\frac{s+8}{s^{3} -3s^{2}+3s-1}$.\nFind the partial fraction decomposition of $F(s)$:\n$ \\frac{s+8}{s^{3} -3s^{2}+3s-1}=$ [ANS]+[ANS]\nFind the inverse Laplace transform of $ F(s)$.\n$ f(t)={\\mathcal L}^{-1} \\left\\lbrace F(s) \\right\\rbrace=$ [ANS]", "answer_v2": [ "9/[(s-1)^3]", "1/[(s-1)^2]", "9/2*e^t*t^2+1*e^t*t" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Consider the function $ F(s)= \\frac{2s+6}{s^{3} -3s^{2}+3s-1}$.\nFind the partial fraction decomposition of $F(s)$:\n$ \\frac{2s+6}{s^{3} -3s^{2}+3s-1}=$ [ANS]+[ANS]\nFind the inverse Laplace transform of $ F(s)$.\n$ f(t)={\\mathcal L}^{-1} \\left\\lbrace F(s) \\right\\rbrace=$ [ANS]", "answer_v3": [ "8/[(s-1)^3]", "2/[(s-1)^2]", "8/2*e^t*t^2+2*e^t*t" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0245", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Inverse transformations", "level": "2", "keywords": [ "Laplace transform" ], "problem_v1": "Find the inverse Laplace transform $f(t)={\\mathcal L}^{-1} \\left\\lbrace F(s) \\right\\rbrace$ of the function $ F(s)= \\frac{5}{s^{2} +49}+ \\frac{2s}{s^{2} +64}$.\n$ f(t)={\\mathcal L}^{-1} \\left\\lbrace \\frac{5}{s^{2} +49}+ \\frac{2s}{s^{2} +64} \\right\\rbrace=$ [ANS]", "answer_v1": [ "0.714286*sin(7*t)+2*cos(8*t)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the inverse Laplace transform $f(t)={\\mathcal L}^{-1} \\left\\lbrace F(s) \\right\\rbrace$ of the function $ F(s)= \\frac{8s}{s^{2} +16}- \\frac{8}{s^{2} +4}$.\n$ f(t)={\\mathcal L}^{-1} \\left\\lbrace \\frac{8s}{s^{2} +16}- \\frac{8}{s^{2} +4} \\right\\rbrace=$ [ANS]", "answer_v2": [ "8*cos(4*t)-4*sin(2*t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the inverse Laplace transform $f(t)={\\mathcal L}^{-1} \\left\\lbrace F(s) \\right\\rbrace$ of the function $ F(s)= \\frac{2s}{s^{2} +36}- \\frac{4}{s^{2} +9}$.\n$ f(t)={\\mathcal L}^{-1} \\left\\lbrace \\frac{2s}{s^{2} +36}- \\frac{4}{s^{2} +9} \\right\\rbrace=$ [ANS]", "answer_v3": [ "2*cos(6*t)-1.33333*sin(3*t)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0246", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Convolutions", "level": "3", "keywords": [ "Laplace transform" ], "problem_v1": "For the functions $f(t)=e^{t}$ and $g(t)=e^{-5t}$, defined on $0\\leq t < \\infty$, compute $f \\ast g$ in two different ways:\nBy directly evaluating the integral in the definition of $f \\ast g$.\n$ (f \\ast g)(t)=\\int_0^t$ [ANS] $dw \\=\\ $ [ANS]\nBy computing ${\\mathcal L}^{-1} \\left\\lbrace F(s)G(s) \\right\\rbrace$ where $F(s)={\\mathcal L} \\left\\lbrace f(t) \\right\\rbrace$ and $G(s)={\\mathcal L} \\left\\lbrace g(t) \\right\\rbrace$.\n$ (f \\ast g)(t)={\\mathcal L}^{-1} \\left\\lbrace F(s)G(s) \\right\\rbrace={\\mathcal L}^{-1} \\big\\lbrace$ [ANS] $\\big\\rbrace$\n$ \\phantom{(f \\ast g)(t)}=$ [ANS]", "answer_v1": [ "e^(t-w)*e^(-5*w)", "e^t/-(1+5)*(e^[-(5+1)*t]-1)", "1/(s-1)*1/(s+5)", "1/(1+5)*[e^t-e^(-5*t)]" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "For the functions $f(t)=e^{t}$ and $g(t)=e^{-2t}$, defined on $0\\leq t < \\infty$, compute $f \\ast g$ in two different ways:\nBy directly evaluating the integral in the definition of $f \\ast g$.\n$ (f \\ast g)(t)=\\int_0^t$ [ANS] $dw \\=\\ $ [ANS]\nBy computing ${\\mathcal L}^{-1} \\left\\lbrace F(s)G(s) \\right\\rbrace$ where $F(s)={\\mathcal L} \\left\\lbrace f(t) \\right\\rbrace$ and $G(s)={\\mathcal L} \\left\\lbrace g(t) \\right\\rbrace$.\n$ (f \\ast g)(t)={\\mathcal L}^{-1} \\left\\lbrace F(s)G(s) \\right\\rbrace={\\mathcal L}^{-1} \\big\\lbrace$ [ANS] $\\big\\rbrace$\n$ \\phantom{(f \\ast g)(t)}=$ [ANS]", "answer_v2": [ "e^(t-w)*e^(-2*w)", "e^t/-(1+2)*(e^[-(2+1)*t]-1)", "1/(s-1)*1/(s+2)", "1/(1+2)*[e^t-e^(-2*t)]" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "For the functions $f(t)=e^{t}$ and $g(t)=e^{-3t}$, defined on $0\\leq t < \\infty$, compute $f \\ast g$ in two different ways:\nBy directly evaluating the integral in the definition of $f \\ast g$.\n$ (f \\ast g)(t)=\\int_0^t$ [ANS] $dw \\=\\ $ [ANS]\nBy computing ${\\mathcal L}^{-1} \\left\\lbrace F(s)G(s) \\right\\rbrace$ where $F(s)={\\mathcal L} \\left\\lbrace f(t) \\right\\rbrace$ and $G(s)={\\mathcal L} \\left\\lbrace g(t) \\right\\rbrace$.\n$ (f \\ast g)(t)={\\mathcal L}^{-1} \\left\\lbrace F(s)G(s) \\right\\rbrace={\\mathcal L}^{-1} \\big\\lbrace$ [ANS] $\\big\\rbrace$\n$ \\phantom{(f \\ast g)(t)}=$ [ANS]", "answer_v3": [ "e^(t-w)*e^(-3*w)", "e^t/-(1+3)*(e^[-(3+1)*t]-1)", "1/(s-1)*1/(s+3)", "1/(1+3)*[e^t-e^{-3*t}]" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0247", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Convolutions", "level": "3", "keywords": [ "Laplace transform" ], "problem_v1": "Consider the following initial value problem, defined for $t \\geq 0$: y'-6 y=\\int_0^t (t-w)\\, e^{6 w} \\ dw, \\quad \\quad y(0)=-4. Find the Laplace transform of the solution.\n$ Y(s)={\\mathcal L}\\left\\lbrace y(t) \\right\\rbrace=$ [ANS]\nObtain the solution $y(t)$.\n$y(t)=$ [ANS]", "answer_v1": [ "1/[s^2*(s-6)^2]-4/(s-6)", "0.0277778*t+0.00925926+0.0277778*e^(6*t)*t-4.00926*e^(6*t)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Consider the following initial value problem, defined for $t \\geq 0$: y'-2 y=\\int_0^t (t-w)\\, e^{2 w} \\ dw, \\quad \\quad y(0)=-1. Find the Laplace transform of the solution.\n$ Y(s)={\\mathcal L}\\left\\lbrace y(t) \\right\\rbrace=$ [ANS]\nObtain the solution $y(t)$.\n$y(t)=$ [ANS]", "answer_v2": [ "1/[s^2*(s-2)^2]-1/(s-2)", "0.25*t+0.25+0.25*e^(2*t)*t-1.25*e^(2*t)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Consider the following initial value problem, defined for $t \\geq 0$: y'-3 y=\\int_0^t (t-w)\\, e^{3 w} \\ dw, \\quad \\quad y(0)=-4. Find the Laplace transform of the solution.\n$ Y(s)={\\mathcal L}\\left\\lbrace y(t) \\right\\rbrace=$ [ANS]\nObtain the solution $y(t)$.\n$y(t)=$ [ANS]", "answer_v3": [ "1/[s^2*(s-3)^2]-4/(s-3)", "0.111111*t+0.0740741+0.111111*e^(3*t)*t-4.07407*e^(3*t)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0248", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Convolutions", "level": "4", "keywords": [], "problem_v1": "Solutions to linear differential equations can be written using convolutions as y \\ \\=\\ \\ y_{\\mathrm{IVP}} \\ \\+\\ \\ \\Bigl(h(t) \\ {\\Large \\ast} \\ f(t)\\Bigr) $\\qquad \\bullet$ $y_{\\mathrm{IVP}}$ is the solution to the associated homogeneous differential equation with the given initial values $\\qquad\\qquad$ (ignore the forcing function, keep initial values). $\\qquad \\bullet$ $h(t)$ is the impulse response $\\qquad\\qquad$ (ignore the initial values and forcing function). $\\qquad \\bullet$ $f(t)$ is the forcing function. $\\qquad\\qquad$ (ignore the initial values and differential equation).\nUse the form above to write the solution to the differential equation y''-4y'+3y=2t^{2}e^{3t} \\qquad \\mathrm{with} \\quad y(0)=-4, \\quad y'(0)=0 $y \\ \\=\\ \\ $ [ANS] $\\quad+\\quad \\Bigl($ [ANS] $\\!\\Large\\ast$ [ANS] $\\Bigr)$\nIf you don't get this in 1 tries, you can get a hint.", "answer_v1": [ "2*e^{3*t}-6*e^t", "[e^{3*t}-e^t]/2", "2*t^2*e^{3*t}" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Solutions to linear differential equations can be written using convolutions as y \\ \\=\\ \\ y_{\\mathrm{IVP}} \\ \\+\\ \\ \\Bigl(h(t) \\ {\\Large \\ast} \\ f(t)\\Bigr) $\\qquad \\bullet$ $y_{\\mathrm{IVP}}$ is the solution to the associated homogeneous differential equation with the given initial values $\\qquad\\qquad$ (ignore the forcing function, keep initial values). $\\qquad \\bullet$ $h(t)$ is the impulse response $\\qquad\\qquad$ (ignore the initial values and forcing function). $\\qquad \\bullet$ $f(t)$ is the forcing function. $\\qquad\\qquad$ (ignore the initial values and differential equation).\nUse the form above to write the solution to the differential equation y''-25y=-7t^{2}e^{-5t} \\qquad \\mathrm{with} \\quad y(0)=-1, \\quad y'(0)=5 $y \\ \\=\\ \\ $ [ANS] $\\quad+\\quad \\Bigl($ [ANS] $\\!\\Large\\ast$ [ANS] $\\Bigr)$\nIf you don't get this in 1 tries, you can get a hint.", "answer_v2": [ "-e^{-5*t}", "-([e^{-5*t}-e^{5*t}]/10)", "-7*t^2*e^{-5*t}" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Solutions to linear differential equations can be written using convolutions as y \\ \\=\\ \\ y_{\\mathrm{IVP}} \\ \\+\\ \\ \\Bigl(h(t) \\ {\\Large \\ast} \\ f(t)\\Bigr) $\\qquad \\bullet$ $y_{\\mathrm{IVP}}$ is the solution to the associated homogeneous differential equation with the given initial values $\\qquad\\qquad$ (ignore the forcing function, keep initial values). $\\qquad \\bullet$ $h(t)$ is the impulse response $\\qquad\\qquad$ (ignore the initial values and forcing function). $\\qquad \\bullet$ $f(t)$ is the forcing function. $\\qquad\\qquad$ (ignore the initial values and differential equation).\nUse the form above to write the solution to the differential equation y''+y'-2y=-4t^{2}e^{-2t} \\qquad \\mathrm{with} \\quad y(0)=-7, \\quad y'(0)=-1 $y \\ \\=\\ \\ $ [ANS] $\\quad+\\quad \\Bigl($ [ANS] $\\!\\Large\\ast$ [ANS] $\\Bigr)$\nIf you don't get this in 1 tries, you can get a hint.", "answer_v3": [ "-[2*e^{-2*t}+5*e^t]", "-([e^{-2*t}-e^t]/3)", "-4*t^2*e^{-2*t}" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0249", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Impulse functions", "level": "4", "keywords": [ "Laplace transform" ], "problem_v1": "Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function. y''+16 \\pi^2 y=4 \\pi \\delta(t-3), \\quad \\quad y(0)=0, \\quad y'(0)=0. Find the Laplace transform of the solution.\n$ Y(s)={\\mathcal L}\\left\\lbrace y(t) \\right\\rbrace=$ [ANS]\nObtain the solution $y(t)$.\n$y(t)=$ [ANS]\nExpress the solution as a piecewise-defined function and think about what happens to the graph of the solution at $t=3$.\n$\\begin{array}{ccc}\\hline y(t)= \\Bigg\\lbrace & & [ANS] \\ \\mbox{if} \\ 0 \\leq t < 3, [ANS] \\ \\mbox{if} \\ 3 \\leq t < \\infty. \\\\ \\hline \\end{array}$", "answer_v1": [ "4*pi*e^(-3*s)/(s^2+16*pi^2)", "h(t-3)*sin(4*pi*(t-3))", "0", "sin(4*pi*(t-3))" ], "answer_type_v1": [ "EX", "EX", "NV", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function. y''+4 \\pi^2 y=2 \\pi \\delta(t-5), \\quad \\quad y(0)=0, \\quad y'(0)=0. Find the Laplace transform of the solution.\n$ Y(s)={\\mathcal L}\\left\\lbrace y(t) \\right\\rbrace=$ [ANS]\nObtain the solution $y(t)$.\n$y(t)=$ [ANS]\nExpress the solution as a piecewise-defined function and think about what happens to the graph of the solution at $t=5$.\n$\\begin{array}{ccc}\\hline y(t)= \\Bigg\\lbrace & & [ANS] \\ \\mbox{if} \\ 0 \\leq t < 5, [ANS] \\ \\mbox{if} \\ 5 \\leq t < \\infty. \\\\ \\hline \\end{array}$", "answer_v2": [ "2*pi*e^(-5*s)/(s^2+4*pi^2)", "h(t-5)*sin(2*pi*(t-5))", "0", "sin(2*pi*(t-5))" ], "answer_type_v2": [ "EX", "EX", "NV", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function. y''+4 \\pi^2 y=2 \\pi \\delta(t-4), \\quad \\quad y(0)=0, \\quad y'(0)=0. Find the Laplace transform of the solution.\n$ Y(s)={\\mathcal L}\\left\\lbrace y(t) \\right\\rbrace=$ [ANS]\nObtain the solution $y(t)$.\n$y(t)=$ [ANS]\nExpress the solution as a piecewise-defined function and think about what happens to the graph of the solution at $t=4$.\n$\\begin{array}{ccc}\\hline y(t)= \\Bigg\\lbrace & & [ANS] \\ \\mbox{if} \\ 0 \\leq t < 4, [ANS] \\ \\mbox{if} \\ 4 \\leq t < \\infty. \\\\ \\hline \\end{array}$", "answer_v3": [ "2*pi*e^(-4*s)/(s^2+4*pi^2)", "h(t-4)*sin(2*pi*(t-4))", "0", "sin(2*pi*(t-4))" ], "answer_type_v3": [ "EX", "EX", "NV", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0250", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Impulse functions", "level": "3", "keywords": [ "Laplace transform" ], "problem_v1": "Sketch the graph of the function k(t)=\\int_0^t \\left(\\delta(w-4)-\\delta(w-7) \\right) dw, \\quad 0\\leq t < \\infty. Use the graph to express $k(t)$ in terms of shifts of the Heaviside step function $h(t)$.\n$ k(t)=$ [ANS]", "answer_v1": [ "h(7-t)-h(4-t)" ], "answer_type_v1": [ "OE" ], "options_v1": [ [] ], "problem_v2": "Sketch the graph of the function k(t)=\\int_0^t \\left(\\delta(w-1)-\\delta(w-6) \\right) dw, \\quad 0\\leq t < \\infty. Use the graph to express $k(t)$ in terms of shifts of the Heaviside step function $h(t)$.\n$ k(t)=$ [ANS]", "answer_v2": [ "h(6-t)-h(1-t)" ], "answer_type_v2": [ "OE" ], "options_v2": [ [] ], "problem_v3": "Sketch the graph of the function k(t)=\\int_0^t \\left(\\delta(w-2)-\\delta(w-6) \\right) dw, \\quad 0\\leq t < \\infty. Use the graph to express $k(t)$ in terms of shifts of the Heaviside step function $h(t)$.\n$ k(t)=$ [ANS]", "answer_v3": [ "h(6-t)-h(2-t)" ], "answer_type_v3": [ "OE" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0251", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Impulse functions", "level": "4", "keywords": [ "Laplace transform" ], "problem_v1": "Let $g(t)=e^{7t}.$\nSolve the initial value problem y'-7 y=g(t), \\quad \\quad y(0)=0, using the technique of integrating factors. (Do not use Laplace transforms.)\n$ y(t)=$ [ANS]\nUse Laplace transforms to determine the transfer function $\\phi(t)$ given the initial value problem \\phi'-7 \\phi=\\delta(t), \\quad \\quad \\phi(0)=0. $ \\phi(t)=$ [ANS]\nEvaluate the convolution integral $(\\phi \\ast g)(t)=\\int_0^t \\phi(t-w)\\, g(w)\\ dw$, and compare the resulting function with the solution obtained in part (a).\n$ (\\phi \\ast g)(t)=\\int_0^t$ [ANS] $dw \\=\\ $ [ANS]", "answer_v1": [ "t*e^(7*t)", "e^(7*t)", "e^[7*(t-w)]*e^(7*w)", "t*e^(7*t)" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $g(t)=e^{2t}.$\nSolve the initial value problem y'-2 y=g(t), \\quad \\quad y(0)=0, using the technique of integrating factors. (Do not use Laplace transforms.)\n$ y(t)=$ [ANS]\nUse Laplace transforms to determine the transfer function $\\phi(t)$ given the initial value problem \\phi'-2 \\phi=\\delta(t), \\quad \\quad \\phi(0)=0. $ \\phi(t)=$ [ANS]\nEvaluate the convolution integral $(\\phi \\ast g)(t)=\\int_0^t \\phi(t-w)\\, g(w)\\ dw$, and compare the resulting function with the solution obtained in part (a).\n$ (\\phi \\ast g)(t)=\\int_0^t$ [ANS] $dw \\=\\ $ [ANS]", "answer_v2": [ "t*e^(2*t)", "e^(2*t)", "e^[2*(t-w)]*e^(2*w)", "t*e^(2*t)" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $g(t)=e^{4t}.$\nSolve the initial value problem y'-4 y=g(t), \\quad \\quad y(0)=0, using the technique of integrating factors. (Do not use Laplace transforms.)\n$ y(t)=$ [ANS]\nUse Laplace transforms to determine the transfer function $\\phi(t)$ given the initial value problem \\phi'-4 \\phi=\\delta(t), \\quad \\quad \\phi(0)=0. $ \\phi(t)=$ [ANS]\nEvaluate the convolution integral $(\\phi \\ast g)(t)=\\int_0^t \\phi(t-w)\\, g(w)\\ dw$, and compare the resulting function with the solution obtained in part (a).\n$ (\\phi \\ast g)(t)=\\int_0^t$ [ANS] $dw \\=\\ $ [ANS]", "answer_v3": [ "t*e^(4*t)", "e^(4*t)", "e^[4*(t-w)]*e^(4*w)", "t*e^(4*t)" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0252", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Impulse functions", "level": "2", "keywords": [ "Laplace transform" ], "problem_v1": "Evaluate the following:\n$ \\int_{-1}^6 (6+e^{-5t})\\, \\delta(t-4) \\ dt=$ [ANS]\n$ \\int_{-1}^6 (6+e^{-5t})\\, \\delta(t-9) \\ dt=$ [ANS]\n$ \\int_{-1}^6 (6+e^{-5t})\\, \\delta(t) \\ dt=$ [ANS]", "answer_v1": [ "6", "0", "7" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Evaluate the following:\n$ \\int_{-1}^6 (9+e^{-t})\\, \\delta(t-2) \\ dt=$ [ANS]\n$ \\int_{-1}^6 (9+e^{-t})\\, \\delta(t-8) \\ dt=$ [ANS]\n$ \\int_{-1}^6 (9+e^{-t})\\, \\delta(t) \\ dt=$ [ANS]", "answer_v2": [ "9.13534", "0", "10" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Evaluate the following:\n$ \\int_{-1}^6 (6+e^{-2t})\\, \\delta(t-3) \\ dt=$ [ANS]\n$ \\int_{-1}^6 (6+e^{-2t})\\, \\delta(t-9) \\ dt=$ [ANS]\n$ \\int_{-1}^6 (6+e^{-2t})\\, \\delta(t) \\ dt=$ [ANS]", "answer_v3": [ "6.00248", "0", "7" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0253", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Step functions", "level": "3", "keywords": [ "laplace", "transform" ], "problem_v1": "Compute the Laplace transform of $f(t)=\\begin{cases} 6, & \\mbox{if} 0\\leq x < 16\\\\ 6 t, & \\mbox{if} 16\\leq x < \\infty \\end{cases}$\n$\\begin{array}{ccccccccccccc}\\hline L\\lbrace f\\rbrace(s)=& & [ANS] \\int [ANS] & & [ANS] & & dt+& & [ANS] \\int [ANS] & & [ANS] & & dt \\\\ \\hline \\end{array}$\n$ u=$ [ANS] $dv=$ [ANS]\n$ du=$ [ANS] $v=$ [ANS]\n$\\begin{array}{ccccccccccccccccccccccc}\\hline =& & [ANS] & & \\Bigg\\vert & & [ANS] [ANS] & &-& & [ANS] & & \\Bigg\\vert & & [ANS] [ANS] & &+& & [ANS] \\int [ANS] & & [ANS] & & dt \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccc}\\hline =& & [ANS] & &-& & [ANS] & & \\Bigg\\vert & & [ANS] [ANS] \\\\ \\hline \\end{array}$\n$=$ [ANS]", "answer_v1": [ "0", "16", "6*e^{-s*t}", "16", "infinity", "6*t*e^{-s*t}", "6*t", "e^{-s*t}*dt", "6*dt", "-[e^{-s*t}/s]", "-6/s*e^{-s*t}", "0", "16", "6*t*e^{-s*t}/s", "16", "infinity", "16", "infinity", "6*e^{-s*t}/s", "6/s+e^(-16*s)*(-6/s+96/s)", "6*e^{-s*t}/(s^2)", "16", "infinity", "6/s+e^(-16*s)*[-6/s+96/s+6/(s^2)]" ], "answer_type_v1": [ "NV", "NV", "EX", "NV", "NV", "EX", "NV", "EX", "EX", "EX", "EX", "NV", "NV", "EX", "NV", "NV", "NV", "NV", "EX", "EX", "EX", "NV", "NV", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Compute the Laplace transform of $f(t)=\\begin{cases} 9, & \\mbox{if} 0\\leq x < 3\\\\ 3 t, & \\mbox{if} 3\\leq x < \\infty \\end{cases}$\n$\\begin{array}{ccccccccccccc}\\hline L\\lbrace f\\rbrace(s)=& & [ANS] \\int [ANS] & & [ANS] & & dt+& & [ANS] \\int [ANS] & & [ANS] & & dt \\\\ \\hline \\end{array}$\n$ u=$ [ANS] $dv=$ [ANS]\n$ du=$ [ANS] $v=$ [ANS]\n$\\begin{array}{ccccccccccccccccccccccc}\\hline =& & [ANS] & & \\Bigg\\vert & & [ANS] [ANS] & &-& & [ANS] & & \\Bigg\\vert & & [ANS] [ANS] & &+& & [ANS] \\int [ANS] & & [ANS] & & dt \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccc}\\hline =& & [ANS] & &-& & [ANS] & & \\Bigg\\vert & & [ANS] [ANS] \\\\ \\hline \\end{array}$\n$=$ [ANS]", "answer_v2": [ "0", "3", "9*e^{-s*t}", "3", "infinity", "3*t*e^{-s*t}", "3*t", "e^{-s*t}*dt", "3*dt", "-[e^{-s*t}/s]", "-9/s*e^{-s*t}", "0", "3", "3*t*e^{-s*t}/s", "3", "infinity", "3", "infinity", "3*e^{-s*t}/s", "9/s+e^(-3*s)*(-9/s+9/s)", "3*e^{-s*t}/(s^2)", "3", "infinity", "9/s+e^(-3*s)*[-9/s+9/s+3/(s^2)]" ], "answer_type_v2": [ "NV", "NV", "EX", "NV", "NV", "EX", "NV", "EX", "EX", "EX", "EX", "NV", "NV", "EX", "NV", "NV", "NV", "NV", "EX", "EX", "EX", "NV", "NV", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Compute the Laplace transform of $f(t)=\\begin{cases} 6, & \\mbox{if} 0\\leq x < 7\\\\ 4 t, & \\mbox{if} 7\\leq x < \\infty \\end{cases}$\n$\\begin{array}{ccccccccccccc}\\hline L\\lbrace f\\rbrace(s)=& & [ANS] \\int [ANS] & & [ANS] & & dt+& & [ANS] \\int [ANS] & & [ANS] & & dt \\\\ \\hline \\end{array}$\n$ u=$ [ANS] $dv=$ [ANS]\n$ du=$ [ANS] $v=$ [ANS]\n$\\begin{array}{ccccccccccccccccccccccc}\\hline =& & [ANS] & & \\Bigg\\vert & & [ANS] [ANS] & &-& & [ANS] & & \\Bigg\\vert & & [ANS] [ANS] & &+& & [ANS] \\int [ANS] & & [ANS] & & dt \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccc}\\hline =& & [ANS] & &-& & [ANS] & & \\Bigg\\vert & & [ANS] [ANS] \\\\ \\hline \\end{array}$\n$=$ [ANS]", "answer_v3": [ "0", "7", "6*e^{-s*t}", "7", "infinity", "4*t*e^{-s*t}", "4*t", "e^{-s*t}*dt", "4*dt", "-[e^{-s*t}/s]", "-6/s*e^{-s*t}", "0", "7", "4*t*e^{-s*t}/s", "7", "infinity", "7", "infinity", "4*e^{-s*t}/s", "6/s+e^(-7*s)*(-6/s+28/s)", "4*e^{-s*t}/(s^2)", "7", "infinity", "6/s+e^(-7*s)*[-6/s+28/s+4/(s^2)]" ], "answer_type_v3": [ "NV", "NV", "EX", "NV", "NV", "EX", "NV", "EX", "EX", "EX", "EX", "NV", "NV", "EX", "NV", "NV", "NV", "NV", "EX", "EX", "EX", "NV", "NV", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Differential_equations_0254", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Step functions", "level": "2", "keywords": [ "integrals", "integration by parts" ], "problem_v1": "Let $f=2^{6x}u(x-7)$,where $u$ is the Heaviside function. Compute $f(-4)=$ [ANS]\n$f(7)=$ [ANS]\n$f(13)=$ [ANS]", "answer_v1": [ "0", "0", "3.02231E+23" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $f=2^{8x}u(x-0)$,where $u$ is the Heaviside function. Compute $f(-7)=$ [ANS]\n$f(0)=$ [ANS]\n$f(8)=$ [ANS]", "answer_v2": [ "0", "0", "1.84467E+19" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $f=2^{6x}u(x-3)$,where $u$ is the Heaviside function. Compute $f(-6)=$ [ANS]\n$f(3)=$ [ANS]\n$f(9)=$ [ANS]", "answer_v3": [ "0", "0", "1.80144E+16" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0255", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Step functions", "level": "3", "keywords": [ "Laplace transform" ], "problem_v1": "Consider the function f(t)=\\left\\lbrace \\begin{array}{l} 0 & \\mbox{if} 0\\leq t < 8\\pi \\\\ \\sin(t-8\\pi) & \\mbox{if} 8\\pi \\leq t. \\end{array} \\right.\nUse the graph of this function to write it in terms of the Heaviside function. Use $h(t-a)$ for the Heaviside function shifted $a$ units horizontally.\n$f(t)=$ [ANS]\nFind the Laplace transform $F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace$.\n$ F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace=$ [ANS]", "answer_v1": [ "h(t-8*pi)*sin(t-8*pi)", "e^(-8*pi*s)/(s^2+1)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Consider the function f(t)=\\left\\lbrace \\begin{array}{l} 0 & \\mbox{if} 0\\leq t < 2\\pi \\\\ \\sin(t-2\\pi) & \\mbox{if} 2\\pi \\leq t. \\end{array} \\right.\nUse the graph of this function to write it in terms of the Heaviside function. Use $h(t-a)$ for the Heaviside function shifted $a$ units horizontally.\n$f(t)=$ [ANS]\nFind the Laplace transform $F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace$.\n$ F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace=$ [ANS]", "answer_v2": [ "h(t-2*pi)*sin(t-2*pi)", "e^(-2*pi*s)/(s^2+1)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Consider the function f(t)=\\left\\lbrace \\begin{array}{l} 0 & \\mbox{if} 0\\leq t < 4\\pi \\\\ \\sin(t-4\\pi) & \\mbox{if} 4\\pi \\leq t. \\end{array} \\right.\nUse the graph of this function to write it in terms of the Heaviside function. Use $h(t-a)$ for the Heaviside function shifted $a$ units horizontally.\n$f(t)=$ [ANS]\nFind the Laplace transform $F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace$.\n$ F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace=$ [ANS]", "answer_v3": [ "h(t-4*pi)*sin(t-4*pi)", "e^(-4*pi*s)/(s^2+1)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0256", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Step functions", "level": "4", "keywords": [ "Laplace transform" ], "problem_v1": "Consider the periodic function $f(t)$ defined as follows: f(t)=5-4e^{-3t} \\mbox {for} 0\\leq t < 3, \\mbox{and} f(t+3)=f(t). Sketch a graph of $f(t)$ over several periods and compute its Laplace transform.\n$ F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace=$ [ANS] $ \\cdot \\int_A^B$ [ANS] $\\ \\ $ where $A=$ [ANS] and $B=$ [ANS]\n$ \\phantom{F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace}=$ [ANS]", "answer_v1": [ "1/[1-e^(-3*s)]", "[5-4*e^{-3*t}]*e^{-s*t}*dt", "0", "3", "1/[1-e^(-3*s)]*([-e^(-3*s)+1]/s*5+(e^[-(s+3)*3]-1)/(s+3)*4)" ], "answer_type_v1": [ "EX", "EX", "NV", "NV", "EX" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Consider the periodic function $f(t)$ defined as follows: f(t)=2-e^{-4t} \\mbox {for} 0\\leq t < 2, \\mbox{and} f(t+2)=f(t). Sketch a graph of $f(t)$ over several periods and compute its Laplace transform.\n$ F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace=$ [ANS] $ \\cdot \\int_A^B$ [ANS] $\\ \\ $ where $A=$ [ANS] and $B=$ [ANS]\n$ \\phantom{F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace}=$ [ANS]", "answer_v2": [ "1/[1-e^(-2*s)]", "[2-e^{-4*t}]*e^{-s*t}*dt", "0", "2", "1/[1-e^(-2*s)]*([-e^(-2*s)+1]/s*2+(e^[-(s+4)*2]-1)/(s+4)*1)" ], "answer_type_v2": [ "EX", "EX", "NV", "NV", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Consider the periodic function $f(t)$ defined as follows: f(t)=3-2e^{-3t} \\mbox {for} 0\\leq t < 2, \\mbox{and} f(t+2)=f(t). Sketch a graph of $f(t)$ over several periods and compute its Laplace transform.\n$ F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace=$ [ANS] $ \\cdot \\int_A^B$ [ANS] $\\ \\ $ where $A=$ [ANS] and $B=$ [ANS]\n$ \\phantom{F(s)={\\mathcal L}\\left\\lbrace f(t) \\right\\rbrace}=$ [ANS]", "answer_v3": [ "1/[1-e^(-2*s)]", "[3-2*e^{-3*t}]*e^{-s*t}*dt", "0", "2", "1/[1-e^(-2*s)]*([-e^(-2*s)+1]/s*3+(e^[-(s+3)*2]-1)/(s+3)*2)" ], "answer_type_v3": [ "EX", "EX", "NV", "NV", "EX" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Differential_equations_0257", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Step functions", "level": "2", "keywords": [ "Laplace transform" ], "problem_v1": "Graph the function $f(t)=5t\\!\\left(h\\!\\left(t-4\\right)-h\\!\\left(t-8\\right)\\right)$ for $0 \\leq t < \\infty$. Use your graph to write this function piecewise as follows:\n$\\begin{array}{ccc}\\hline 5t\\!\\left(h\\!\\left(t-4\\right)-h\\!\\left(t-8\\right)\\right)= \\Bigg\\lbrace & & [ANS]if 0 \\leq t < 4, [ANS]if 4 \\leq t < 8, [ANS]if 8 \\leq t < \\infty. \\\\ \\hline \\end{array}$\nEvaluate $f(6)$.\n$ f(6)=$ [ANS]", "answer_v1": [ "0", "5*t", "0", "5*6" ], "answer_type_v1": [ "NV", "EX", "NV", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Graph the function $f(t)=2t\\!\\left(h\\!\\left(t-1\\right)-h\\!\\left(t-10\\right)\\right)$ for $0 \\leq t < \\infty$. Use your graph to write this function piecewise as follows:\n$\\begin{array}{ccc}\\hline 2t\\!\\left(h\\!\\left(t-1\\right)-h\\!\\left(t-10\\right)\\right)= \\Bigg\\lbrace & & [ANS]if 0 \\leq t < 1, [ANS]if 1 \\leq t < 10, [ANS]if 10 \\leq t < \\infty. \\\\ \\hline \\end{array}$\nEvaluate $f(5.5)$.\n$ f(5.5)=$ [ANS]", "answer_v2": [ "0", "2*t", "0", "2*5.5" ], "answer_type_v2": [ "NV", "EX", "NV", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Graph the function $f(t)=3t\\!\\left(h\\!\\left(t-2\\right)-h\\!\\left(t-9\\right)\\right)$ for $0 \\leq t < \\infty$. Use your graph to write this function piecewise as follows:\n$\\begin{array}{ccc}\\hline 3t\\!\\left(h\\!\\left(t-2\\right)-h\\!\\left(t-9\\right)\\right)= \\Bigg\\lbrace & & [ANS]if 0 \\leq t < 2, [ANS]if 2 \\leq t < 9, [ANS]if 9 \\leq t < \\infty. \\\\ \\hline \\end{array}$\nEvaluate $f(5.5)$.\n$ f(5.5)=$ [ANS]", "answer_v3": [ "0", "3*t", "0", "3*5.5" ], "answer_type_v3": [ "NV", "EX", "NV", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0258", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Step functions", "level": "4", "keywords": [], "problem_v1": "Find the Laplace transform of f(t)=\\begin{cases} 0, & t < 5 \\cr t^2-10 t+31, & t \\geq 5 \\end{cases} $F(s)=$ [ANS].", "answer_v1": [ "exp(-5*s)*(2/s^3 + 6/s)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the Laplace transform of f(t)=\\begin{cases} 0, & t < 2 \\cr t^2-4 t+13, & t \\geq 2 \\end{cases} $F(s)=$ [ANS].", "answer_v2": [ "exp(-2*s)*(2/s^3 + 9/s)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the Laplace transform of f(t)=\\begin{cases} 0, & t < 3 \\cr t^2-6 t+15, & t \\geq 3 \\end{cases} $F(s)=$ [ANS].", "answer_v3": [ "exp(-3*s)*(2/s^3 + 6/s)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0259", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Step functions", "level": "4", "keywords": [ "Laplace transform" ], "problem_v1": "Compute the inverse Laplace transform: $ \\mathcal{L}^{\\text{-}1}\\,\\biggl\\lbrace \\frac{s-3}{s^{2} -4s+5}e^{-5s} \\biggr\\rbrace=$ [ANS]\n(Notation: write u(t-c) for the Heaviside step function $u_c(t)$ with step at $t=c$.)\nIf you don't get this in 2 tries, you can get a hint.", "answer_v1": [ "u(t-5)*[cos(t-5)-sin(t-5)]*e^[2*(t-5)]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Compute the inverse Laplace transform: $ \\mathcal{L}^{\\text{-}1}\\,\\biggl\\lbrace \\frac{4-s}{s^{2} +8s+32}e^{-2s} \\biggr\\rbrace=$ [ANS]\n(Notation: write u(t-c) for the Heaviside step function $u_c(t)$ with step at $t=c$.)\nIf you don't get this in 2 tries, you can get a hint.", "answer_v2": [ "u(t-2)*[2*sin(4*(t-2))-cos(4*(t-2))]*e^[-4*(t-2)]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Compute the inverse Laplace transform: $ \\mathcal{L}^{\\text{-}1}\\,\\biggl\\lbrace \\frac{-s-3}{s^{2} +4s+5}e^{-3s} \\biggr\\rbrace=$ [ANS]\n(Notation: write u(t-c) for the Heaviside step function $u_c(t)$ with step at $t=c$.)\nIf you don't get this in 2 tries, you can get a hint.", "answer_v3": [ "-u(t-3)*[cos(t-3)+sin(t-3)]*e^[-2*(t-3)]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0260", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Step functions", "level": "4", "keywords": [ "Laplace transform" ], "problem_v1": "Compute the Laplace transform. Your answer should be a function of the variable $s$: $ \\mathcal{L}\\,\\biggl\\lbrace 5+u_{3/2}(t)e^{t}\\sin\\!\\left(\\pi t\\right) \\biggr\\rbrace=$ [ANS]\nYou may find the following formulas useful: $\\qquad\\begin{aligned}\\cos(bt+\\pi) &=-\\cos(bt) \\\\ \\sin(bt+\\pi) &=-\\sin(bt) \\\\ \\cos(bt+\\tfrac{\\pi}{2}) &=-\\sin(bt) \\\\ \\sin(bt+\\tfrac{\\pi}{2}) &=\\cos(bt) \\\\ \\end{aligned}$ If you don't get this in 2 tries, you can get a hint.", "answer_v1": [ "5/s+[-(s-1)]/[(s-1)^2+pi^2]*e^[-1.5*(s-1)]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Compute the Laplace transform. Your answer should be a function of the variable $s$: $ \\mathcal{L}\\,\\biggl\\lbrace 1+u_{3}(t)e^{6t}\\cos\\!\\left(\\pi t\\right) \\biggr\\rbrace=$ [ANS]\nYou may find the following formulas useful: $\\qquad\\begin{aligned}\\cos(bt+\\pi) &=-\\cos(bt) \\\\ \\sin(bt+\\pi) &=-\\sin(bt) \\\\ \\cos(bt+\\tfrac{\\pi}{2}) &=-\\sin(bt) \\\\ \\sin(bt+\\tfrac{\\pi}{2}) &=\\cos(bt) \\\\ \\end{aligned}$ If you don't get this in 2 tries, you can get a hint.", "answer_v2": [ "1/s+[-(s-6)]/[(s-6)^2+pi^2]*e^[-3*(s-6)]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Compute the Laplace transform. Your answer should be a function of the variable $s$: $ \\mathcal{L}\\,\\biggl\\lbrace 2+u_{3/2}(t)e^{t}\\cos\\!\\left(\\pi t\\right) \\biggr\\rbrace=$ [ANS]\nYou may find the following formulas useful: $\\qquad\\begin{aligned}\\cos(bt+\\pi) &=-\\cos(bt) \\\\ \\sin(bt+\\pi) &=-\\sin(bt) \\\\ \\cos(bt+\\tfrac{\\pi}{2}) &=-\\sin(bt) \\\\ \\sin(bt+\\tfrac{\\pi}{2}) &=\\cos(bt) \\\\ \\end{aligned}$ If you don't get this in 2 tries, you can get a hint.", "answer_v3": [ "2/s+pi/[(s-1)^2+pi^2]*e^[-1.5*(s-1)]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0261", "subject": "Differential_equations", "topic": "Laplace transforms", "subtopic": "Step functions", "level": "4", "keywords": [ "Laplace transform" ], "problem_v1": "Compute the Laplace transform: $ \\mathcal{L}\\,\\biggl\\lbrace u_4(t)+u_5(t)t^{2}e^{3t} \\biggr\\rbrace=$ [ANS]\nIf you don't get this in 2 tries, you can get a hint.", "answer_v1": [ "e^[-5*(s-3)]*(2/[(s-3)^3]+10/[(s-3)^2]+25/(s-3))+e^(-4*s)/s" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Compute the Laplace transform: $ \\mathcal{L}\\,\\biggl\\lbrace u_1(t)te^{-2t}+u_6(t) \\biggr\\rbrace=$ [ANS]\nIf you don't get this in 2 tries, you can get a hint.", "answer_v2": [ "e^[-(s+2)]*(1/[(s+2)^2]+1/(s+2))+e^(-6*s)/s" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Compute the Laplace transform: $ \\mathcal{L}\\,\\biggl\\lbrace u_2(t)te^{t}+u_4(t) \\biggr\\rbrace=$ [ANS]\nIf you don't get this in 2 tries, you can get a hint.", "answer_v3": [ "e^[-2*(s-1)]*(1/[(s-1)^2]+2/(s-1))+e^(-4*s)/s" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0262", "subject": "Differential_equations", "topic": "Systems of differential equations", "subtopic": "Applications", "level": "2", "keywords": [ "differential equation' 'first order' 'model" ], "problem_v1": "Consider the following model for the populations of rabbits and wolves (where $R$ is the population of rabbits and $W$ is the population of wolves).\n\\begin{array}{rcl} \\frac{dR}{dt} &=& 0.05 R (1-0.00032 R)-0.000525 RW \\cr & & \\cr \\frac{dW}{dt} &=&-0.05 W+0.0001 RW \\end{array} Find all the equilibrium solutions:\n(a) In the absence of wolves, the population of rabbits approaches [ANS]. (b) In the absence of rabbits, the population of wolves approaches [ANS]. (c) If both wolves and rabbits are present, their populations approach $r=$ [ANS] and $w=$ [ANS].", "answer_v1": [ "3125", "0", "500", "80" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the following model for the populations of rabbits and wolves (where $R$ is the population of rabbits and $W$ is the population of wolves).\n\\begin{array}{rcl} \\frac{dR}{dt} &=& 0.08 R (1-0.001 R)-0.00125 RW \\cr & & \\cr \\frac{dW}{dt} &=&-0.02 W+4\\times 10^{-5} RW \\end{array} Find all the equilibrium solutions:\n(a) In the absence of wolves, the population of rabbits approaches [ANS]. (b) In the absence of rabbits, the population of wolves approaches [ANS]. (c) If both wolves and rabbits are present, their populations approach $r=$ [ANS] and $w=$ [ANS].", "answer_v2": [ "1000", "0", "500", "32" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the following model for the populations of rabbits and wolves (where $R$ is the population of rabbits and $W$ is the population of wolves).\n\\begin{array}{rcl} \\frac{dR}{dt} &=& 0.1 R (1-0.0005 R)-0.002 RW \\cr & & \\cr \\frac{dW}{dt} &=&-0.04 W+0.0001 RW \\end{array} Find all the equilibrium solutions:\n(a) In the absence of wolves, the population of rabbits approaches [ANS]. (b) In the absence of rabbits, the population of wolves approaches [ANS]. (c) If both wolves and rabbits are present, their populations approach $r=$ [ANS] and $w=$ [ANS].", "answer_v3": [ "2000", "0", "400", "40" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0263", "subject": "Differential_equations", "topic": "Systems of differential equations", "subtopic": "Applications", "level": "4", "keywords": [ "differential equation' 'first order' 'matrices" ], "problem_v1": "Liam opens a bank account with an initial balance of 1500 dollars. Let $b(t)$ be the balance in the account at time $t$. Thus $b(0)=1500$. The bank is paying interest at a continuous rate of 5\\% per year. Liam makes deposits into the account at a continuous rate of $s(t)$ dollars per year. Suppose that $s(0)=1200$ and that $s(t)$ is increasing at a continuous rate of 3\\% per year (Liam can save more as his income goes up over time).\n(a) Set up a linear system of the form \\begin{array}{rcl} \\frac{db}{dt} &=& m_{11} b+m_{12} s, \\cr & & \\cr \\frac{ds}{dt} &=& m_{21} b+m_{22} s. \\end{array} $m_{11}=$ [ANS], $m_{12}=$ [ANS], $m_{21}=$ [ANS], $m_{22}=$ [ANS].\n(b) Find $b(t)$ and $s(t)$. $b(t)=$ [ANS], $s(t)=$ [ANS].", "answer_v1": [ "0.05", "1", "0", "0.03", "(1500 - 1200/(0.03 - 0.05)) * e^(0.05*t) + 1200/(0.03 - 0.05) * e^(0.03*t)", "1200 * e^(0.03*t)" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Mike opens a bank account with an initial balance of 2000 dollars. Let $b(t)$ be the balance in the account at time $t$. Thus $b(0)=2000$. The bank is paying interest at a continuous rate of 3\\% per year. Mike makes deposits into the account at a continuous rate of $s(t)$ dollars per year. Suppose that $s(0)=1000$ and that $s(t)$ is increasing at a continuous rate of 6\\% per year (Mike can save more as his income goes up over time).\n(a) Set up a linear system of the form \\begin{array}{rcl} \\frac{db}{dt} &=& m_{11} b+m_{12} s, \\cr & & \\cr \\frac{ds}{dt} &=& m_{21} b+m_{22} s. \\end{array} $m_{11}=$ [ANS], $m_{12}=$ [ANS], $m_{21}=$ [ANS], $m_{22}=$ [ANS].\n(b) Find $b(t)$ and $s(t)$. $b(t)=$ [ANS], $s(t)=$ [ANS].", "answer_v2": [ "0.03", "1", "0", "0.06", "(2000 - 1000/(0.06 - 0.03)) * e^(0.03*t) + 1000/(0.06 - 0.03) * e^(0.06*t)", "1000 * e^(0.06*t)" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "John opens a bank account with an initial balance of 1500 dollars. Let $b(t)$ be the balance in the account at time $t$. Thus $b(0)=1500$. The bank is paying interest at a continuous rate of 4\\% per year. John makes deposits into the account at a continuous rate of $s(t)$ dollars per year. Suppose that $s(0)=1100$ and that $s(t)$ is increasing at a continuous rate of 3\\% per year (John can save more as his income goes up over time).\n(a) Set up a linear system of the form \\begin{array}{rcl} \\frac{db}{dt} &=& m_{11} b+m_{12} s, \\cr & & \\cr \\frac{ds}{dt} &=& m_{21} b+m_{22} s. \\end{array} $m_{11}=$ [ANS], $m_{12}=$ [ANS], $m_{21}=$ [ANS], $m_{22}=$ [ANS].\n(b) Find $b(t)$ and $s(t)$. $b(t)=$ [ANS], $s(t)=$ [ANS].", "answer_v3": [ "0.04", "1", "0", "0.03", "(1500 - 1100/(0.03 - 0.04)) * e^(0.04*t) + 1100/(0.03 - 0.04) * e^(0.03*t)", "1100 * e^(0.03*t)" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Differential_equations_0265", "subject": "Differential_equations", "topic": "Systems of differential equations", "subtopic": "Applications", "level": "5", "keywords": [], "problem_v1": "Consider the interaction of two species of animals in a habitat. We are told that the change of the populations $x(t)$ and $y(t)$ can be modeled by the equations \\begin{array}{rcl} \\frac{dx}{dt} &=& 0.4x+1.5 y, \\cr & & \\cr \\frac{dy}{dt} &=& 0.5x-0.6 y. \\end{array} For this system, the smaller eigenvalue is [ANS] and the larger eigenvalue is [ANS].\n[ANS] 1. What kind of interaction do we observe?\n[Note--you may want to view a (right click to open in a new window).] If $y'=Ay$ is a differential equation, how would the solution curves behave? [ANS] A. The solution curves would race towards zero and then veer away towards infinity. (Saddle) B. The solution curves converge to different points. C. All of the solution curves would run away from 0. (Unstable node) D. All of the solutions curves would converge towards 0. (Stable node)\nThe solution to the above differential equation with initial values $x(0)=4, \\,\\, y(0)=4$ is $x(t)=$ [ANS], $y(t)=$ [ANS].", "answer_v1": [ "-1.1", "0.9", "SYMBIOSIS", "A", "-1.33333333333333*1.5*exp(-1.1*t)+4*1.5*exp(0.9*t)", "-1.33333333333333*(-1.1-0.4)*exp(-1.1*t)+4*(0.9-0.4)*exp(0.9*t)" ], "answer_type_v1": [ "NV", "NV", "MCS", "MCS", "EX", "EX" ], "options_v1": [ [], [], [ "Competition", "Predator-prey" ], [ "A", "B", "C", "D" ], [], [] ], "problem_v2": "Consider the interaction of two species of animals in a habitat. We are told that the change of the populations $x(t)$ and $y(t)$ can be modeled by the equations \\begin{array}{rcl} \\frac{dx}{dt} &=& 0.4x+1.5 y, \\cr & & \\cr \\frac{dy}{dt} &=& 0.5x-0.6 y. \\end{array} For this system, the smaller eigenvalue is [ANS] and the larger eigenvalue is [ANS].\n[ANS] 1. What kind of interaction do we observe?\n[Note--you may want to view a (right click to open in a new window).] If $y'=Ay$ is a differential equation, how would the solution curves behave? [ANS] A. All of the solution curves would run away from 0. (Unstable node) B. The solution curves converge to different points. C. All of the solutions curves would converge towards 0. (Stable node) D. The solution curves would race towards zero and then veer away towards infinity. (Saddle)\nThe solution to the above differential equation with initial values $x(0)=9, \\,\\, y(0)=4$ is $x(t)=$ [ANS], $y(t)=$ [ANS].", "answer_v2": [ "-1.1", "0.9", "SYMBIOSIS", "D", "-0.5*1.5*exp(-1.1*t)+6.5*1.5*exp(0.9*t)", "-0.5*(-1.1-0.4)*exp(-1.1*t)+6.5*(0.9-0.4)*exp(0.9*t)" ], "answer_type_v2": [ "NV", "NV", "MCS", "MCS", "EX", "EX" ], "options_v2": [ [], [], [ "Competition", "Predator-prey" ], [ "A", "B", "C", "D" ], [], [] ], "problem_v3": "Consider the interaction of two species of animals in a habitat. We are told that the change of the populations $x(t)$ and $y(t)$ can be modeled by the equations \\begin{array}{rcl} \\frac{dx}{dt} &=& 0.4x+1.5 y, \\cr & & \\cr \\frac{dy}{dt} &=& 0.5x-0.6 y. \\end{array} For this system, the smaller eigenvalue is [ANS] and the larger eigenvalue is [ANS].\n[ANS] 1. What kind of interaction do we observe?\n[Note--you may want to view a (right click to open in a new window).] If $y'=Ay$ is a differential equation, how would the solution curves behave? [ANS] A. The solution curves would race towards zero and then veer away towards infinity. (Saddle) B. All of the solutions curves would converge towards 0. (Stable node) C. All of the solution curves would run away from 0. (Unstable node) D. The solution curves converge to different points.\nThe solution to the above differential equation with initial values $x(0)=3, \\,\\, y(0)=4$ is $x(t)=$ [ANS], $y(t)=$ [ANS].", "answer_v3": [ "-1.1", "0.9", "SYMBIOSIS", "A", "-1.5*1.5*exp(-1.1*t)+3.5*1.5*exp(0.9*t)", "-1.5*(-1.1-0.4)*exp(-1.1*t)+3.5*(0.9-0.4)*exp(0.9*t)" ], "answer_type_v3": [ "NV", "NV", "MCS", "MCS", "EX", "EX" ], "options_v3": [ [], [], [ "Competition", "Predator-prey" ], [ "A", "B", "C", "D" ], [], [] ] }, { "id": "Differential_equations_0266", "subject": "Differential_equations", "topic": "Systems of differential equations", "subtopic": "Distinct real eigenvalues", "level": "3", "keywords": [ "calculus", "integral", "differential equations", "higher derivatives" ], "problem_v1": "Consider the system of differential equations \\frac{dx}{dt} =-4 y \\qquad\\qquad \\frac{dy}{dt} =-4x. Convert this system to a second order differential equation in $y$ by differentiating the second equation with respect to $t$ and substituting for $x$ from the first equation. Solve the equation you obtained for $y$ as a function of $t$ ; hence find $x$ as a function of $t$. If we also require $x(0)=3$ and $y(0)=4$, what are $x$ and $y$? $x(t)=$ [ANS]\n$y(t)=$ [ANS]", "answer_v1": [ "3.5*e^{-4*t}-0.5*e^(4*t)", "3.5*e^{-4*t}+0.5*e^(4*t)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Consider the system of differential equations \\frac{dx}{dt} =-y \\qquad\\qquad \\frac{dy}{dt} =-x. Convert this system to a second order differential equation in $y$ by differentiating the second equation with respect to $t$ and substituting for $x$ from the first equation. Solve the equation you obtained for $y$ as a function of $t$ ; hence find $x$ as a function of $t$. If we also require $x(0)=5$ and $y(0)=1$, what are $x$ and $y$? $x(t)=$ [ANS]\n$y(t)=$ [ANS]", "answer_v2": [ "3*e^(-1*t)--2*e^(1*t)", "3*e^(-1*t)+-2*e^(1*t)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Consider the system of differential equations \\frac{dx}{dt} =-2 y \\qquad\\qquad \\frac{dy}{dt} =-2x. Convert this system to a second order differential equation in $y$ by differentiating the second equation with respect to $t$ and substituting for $x$ from the first equation. Solve the equation you obtained for $y$ as a function of $t$ ; hence find $x$ as a function of $t$. If we also require $x(0)=4$ and $y(0)=2$, what are $x$ and $y$? $x(t)=$ [ANS]\n$y(t)=$ [ANS]", "answer_v3": [ "3*e^(-2*t)--1*e^(2*t)", "3*e^(-2*t)+-1*e^(2*t)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0267", "subject": "Differential_equations", "topic": "Systems of differential equations", "subtopic": "Distinct real eigenvalues", "level": "4", "keywords": [], "problem_v1": "Given that $\\boldsymbol{\\vec{v}_1}=\\left\\lbrack \\begin{array}{r}-1 \\\\-1 \\end{array} \\right\\rbrack$ and $\\boldsymbol{\\vec{v}_2}=\\left\\lbrack \\begin{array}{r} 2 \\\\ 1 \\end{array} \\right\\rbrack$ are eigenvectors of the matrix $\\left\\lbrack \\begin{array}{rr}-6 & 10 \\\\-5 & 9 \\end{array} \\right\\rbrack$, determine the corresponding eigenvalues. $\\lambda_1=$ [ANS]\n$\\lambda_2=$ [ANS]\nFind the solution to the linear system of differential equations $ \\left\\lbrace \\begin{array}{rcl} x^{\\,\\prime} &=&-6x+10y \\\\ y^{\\,\\prime} &=&-5x+9y \\end{array} \\right.$ satisfying the initial conditions $x(0)=3$ and $y(0)=2$.\n$x(t)=$ [ANS]\n$y(t)=$ [ANS]", "answer_v1": [ "4", "-1", "-1*-1*e^(4*t)+1*2*e^(-1*t)", "-1*-1*e^(4*t)+1*1*e^(-1*t)" ], "answer_type_v1": [ "NV", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Given that $\\boldsymbol{\\vec{v}_1}=\\left\\lbrack \\begin{array}{r}-1 \\\\-2 \\end{array} \\right\\rbrack$ and $\\boldsymbol{\\vec{v}_2}=\\left\\lbrack \\begin{array}{r} 0 \\\\-1 \\end{array} \\right\\rbrack$ are eigenvectors of the matrix $\\left\\lbrack \\begin{array}{rr}-1 & 0 \\\\ 10 &-6 \\end{array} \\right\\rbrack$, determine the corresponding eigenvalues. $\\lambda_1=$ [ANS]\n$\\lambda_2=$ [ANS]\nFind the solution to the linear system of differential equations $ \\left\\lbrace \\begin{array}{rcl} x^{\\,\\prime} &=&-x \\\\ y^{\\,\\prime} &=& 10x-6y \\end{array} \\right.$ satisfying the initial conditions $x(0)=3$ and $y(0)=5$.\n$x(t)=$ [ANS]\n$y(t)=$ [ANS]", "answer_v2": [ "-1", "-6", "-3*-1*e^(-1*t)+1*0*e^(-6*t)", "-3*-2*e^(-1*t)+1*-1*e^(-6*t)" ], "answer_type_v2": [ "NV", "NV", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Given that $\\boldsymbol{\\vec{v}_1}=\\left\\lbrack \\begin{array}{r} 1 \\\\-1 \\end{array} \\right\\rbrack$ and $\\boldsymbol{\\vec{v}_2}=\\left\\lbrack \\begin{array}{r} 2 \\\\-1 \\end{array} \\right\\rbrack$ are eigenvectors of the matrix $\\left\\lbrack \\begin{array}{rr} 10 & 14 \\\\-7 &-11 \\end{array} \\right\\rbrack$, determine the corresponding eigenvalues. $\\lambda_1=$ [ANS]\n$\\lambda_2=$ [ANS]\nFind the solution to the linear system of differential equations $ \\left\\lbrace \\begin{array}{rcl} x^{\\,\\prime} &=& 10x+14y \\\\ y^{\\,\\prime} &=&-7x-11y \\end{array} \\right.$ satisfying the initial conditions $x(0)=9$ and $y(0)=-6$.\n$x(t)=$ [ANS]\n$y(t)=$ [ANS]", "answer_v3": [ "-4", "3", "3*1*e^{-4*t}+3*2*e^(3*t)", "3*-1*e^{-4*t}+3*-1*e^(3*t)" ], "answer_type_v3": [ "NV", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0268", "subject": "Differential_equations", "topic": "Systems of differential equations", "subtopic": "Reduction to first order systems", "level": "3", "keywords": [ "differential equations", "systems of ODEs" ], "problem_v1": "Consider the initial value problem\n\\left\\lbrack \\begin{array}{c} y_1^{\\,\\prime} \\\\ y_2^{\\,\\prime} \\end{array} \\right\\rbrack=\\left\\lbrack \\begin{array}{rr} 0 & 1 \\\\ 4 & 5 \\end{array} \\right\\rbrack \\left\\lbrack \\begin{array}{c} y_1 \\\\ y_2 \\end{array} \\right\\rbrack+\\left\\lbrack \\begin{array}{c} 0 \\\\ 3\\cos\\!\\left(4t\\right) \\end{array} \\right\\rbrack,\n\\left\\lbrack \\begin{array}{c} y_1(1) \\\\ y_2(1) \\end{array} \\right\\rbrack=\\left\\lbrack \\begin{array}{r}-1 \\\\ 2 \\end{array} \\right\\rbrack. This initial value problem was obtained from an initial value problem for a higher order scalar differential equation, via the change of variables $y_1=y$ and $y_2=y'$. What is the corresponding scalar initial value problem?\nDifferential equation: [ANS]\n(Give your answer in terms of $y, y^{\\,\\prime}, y^{\\,\\prime\\prime}, t$.)\nInitial conditions: [ANS] and [ANS]\n(Give your first answer in the form $y(t_0)=y_0$.\nGive your second answer in the form $y'(t_0)=y_0'$.)", "answer_v1": [ "y", "y*1 = -1", "y" ], "answer_type_v1": [ "EX", "EQ", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Consider the initial value problem\n\\left\\lbrack \\begin{array}{c} y_1^{\\,\\prime} \\\\ y_2^{\\,\\prime} \\end{array} \\right\\rbrack=\\left\\lbrack \\begin{array}{rr} 0 & 1 \\\\-6 &-3 \\end{array} \\right\\rbrack \\left\\lbrack \\begin{array}{c} y_1 \\\\ y_2 \\end{array} \\right\\rbrack+\\left\\lbrack \\begin{array}{c} 0 \\\\ 5\\cos\\!\\left(3t\\right) \\end{array} \\right\\rbrack,\n\\left\\lbrack \\begin{array}{c} y_1(-3) \\\\ y_2(-3) \\end{array} \\right\\rbrack=\\left\\lbrack \\begin{array}{r}-2 \\\\ 1 \\end{array} \\right\\rbrack. This initial value problem was obtained from an initial value problem for a higher order scalar differential equation, via the change of variables $y_1=y$ and $y_2=y'$. What is the corresponding scalar initial value problem?\nDifferential equation: [ANS]\n(Give your answer in terms of $y, y^{\\,\\prime}, y^{\\,\\prime\\prime}, t$.)\nInitial conditions: [ANS] and [ANS]\n(Give your first answer in the form $y(t_0)=y_0$.\nGive your second answer in the form $y'(t_0)=y_0'$.)", "answer_v2": [ "y", "y*(-3) = -2", "y" ], "answer_type_v2": [ "EX", "EQ", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Consider the initial value problem\n\\left\\lbrack \\begin{array}{c} y_1^{\\,\\prime} \\\\ y_2^{\\,\\prime} \\end{array} \\right\\rbrack=\\left\\lbrack \\begin{array}{rr} 0 & 1 \\\\-5 &-4 \\end{array} \\right\\rbrack \\left\\lbrack \\begin{array}{c} y_1 \\\\ y_2 \\end{array} \\right\\rbrack+\\left\\lbrack \\begin{array}{c} 0 \\\\ 2\\cos\\!\\left(3t\\right) \\end{array} \\right\\rbrack,\n\\left\\lbrack \\begin{array}{c} y_1(3) \\\\ y_2(3) \\end{array} \\right\\rbrack=\\left\\lbrack \\begin{array}{r} 5 \\\\ 4 \\end{array} \\right\\rbrack. This initial value problem was obtained from an initial value problem for a higher order scalar differential equation, via the change of variables $y_1=y$ and $y_2=y'$. What is the corresponding scalar initial value problem?\nDifferential equation: [ANS]\n(Give your answer in terms of $y, y^{\\,\\prime}, y^{\\,\\prime\\prime}, t$.)\nInitial conditions: [ANS] and [ANS]\n(Give your first answer in the form $y(t_0)=y_0$.\nGive your second answer in the form $y'(t_0)=y_0'$.)", "answer_v3": [ "y", "y*3 = 5", "y" ], "answer_type_v3": [ "EX", "EQ", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0269", "subject": "Differential_equations", "topic": "Systems of differential equations", "subtopic": "Nonhomogeneous systems", "level": "3", "keywords": [ "differential equations", "systems of ODEs" ], "problem_v1": "Find the solution to\n\\begin{array}{lcl} x^{\\,\\prime} &=& y-x+t \\\\ y^{\\,\\prime} &=& y \\end{array}\nif $x(0)=7$ and $y(0)=6$.\n$x(t)=$ [ANS]\n$y(t)=$ [ANS]", "answer_v1": [ "5*e^(-t)+3*e^t+t-1", "6*e^t" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Find the solution to\n\\begin{array}{lcl} x^{\\,\\prime} &=& y-x+t \\\\ y^{\\,\\prime} &=& y \\end{array}\nif $x(0)=1$ and $y(0)=8$.\n$x(t)=$ [ANS]\n$y(t)=$ [ANS]", "answer_v2": [ "4*e^t-2*e^(-t)+t-1", "8*e^t" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Find the solution to\n\\begin{array}{lcl} x^{\\,\\prime} &=& y-x+t \\\\ y^{\\,\\prime} &=& y \\end{array}\nif $x(0)=3$ and $y(0)=6$.\n$x(t)=$ [ANS]\n$y(t)=$ [ANS]", "answer_v3": [ "e^(-t)+3*e^t+t-1", "6*e^t" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0274", "subject": "Differential_equations", "topic": "Numerical methods", "subtopic": "Euler", "level": "4", "keywords": [ "calculus", "differential equations", "graph", "numerical", "graphical and numerical methods" ], "problem_v1": "Let $y(t)$ be the solution to $\\dot{y}=8 t e^{-y}$ satisfying $y(0)=1$.\n(a) Use Euler's Method with time step $h=0.2$ to approximate $y(0.2), \\, y(0.4),..., y(1.0)$. $\\begin{array}{ccc}\\hline k & t_k & y_k \\\\ \\hline 0 & 0 & 1 \\\\ \\hline 1 & 0.2 & [ANS] \\\\ \\hline 2 & 0.4 & [ANS] \\\\ \\hline 3 & 0.6 & [ANS] \\\\ \\hline 4 & 0.8 & [ANS] \\\\ \\hline 5 & 1.0 & [ANS] \\\\ \\hline \\end{array}$\n(b) Use separation of variables to find $y(t)$ exactly. $y(t)$=[ANS]\n(c) Compute the error in the approximations to $y(0.2),\\, y(0.6)$, and $y(1)$. $\\left|y(0.2)-y_1\\right|=$ [ANS]\n$\\left|y(0.6)-y_3\\right|=$ [ANS]\n$\\left|y(1)-y_{5}\\right|=$ [ANS]", "answer_v1": [ "1", "1.11772", "1.32702", "1.58167", "1.84488", "ln(4*t^2+2.71828)", "0.0571929", "0.0980847", "0.0599513" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let $y(t)$ be the solution to $\\dot{y}=2 t e^{-y}$ satisfying $y(0)=3$.\n(a) Use Euler's Method with time step $h=0.2$ to approximate $y(0.2), \\, y(0.4),..., y(1.0)$. $\\begin{array}{ccc}\\hline k & t_k & y_k \\\\ \\hline 0 & 0 & 3 \\\\ \\hline 1 & 0.2 & [ANS] \\\\ \\hline 2 & 0.4 & [ANS] \\\\ \\hline 3 & 0.6 & [ANS] \\\\ \\hline 4 & 0.8 & [ANS] \\\\ \\hline 5 & 1.0 & [ANS] \\\\ \\hline \\end{array}$\n(b) Use separation of variables to find $y(t)$ exactly. $y(t)$=[ANS]\n(c) Compute the error in the approximations to $y(0.2),\\, y(0.6)$, and $y(1)$. $\\left|y(0.2)-y_1\\right|=$ [ANS]\n$\\left|y(0.6)-y_3\\right|=$ [ANS]\n$\\left|y(1)-y_{5}\\right|=$ [ANS]", "answer_v2": [ "3", "3.00398", "3.01192", "3.02372", "3.03928", "ln(1*t^2+20.0855)", "0.00198767", "0.00584558", "0.00930269" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let $y(t)$ be the solution to $\\dot{y}=4 t e^{-y}$ satisfying $y(0)=1$.\n(a) Use Euler's Method with time step $h=0.2$ to approximate $y(0.2), \\, y(0.4),..., y(1.0)$. $\\begin{array}{ccc}\\hline k & t_k & y_k \\\\ \\hline 0 & 0 & 1 \\\\ \\hline 1 & 0.2 & [ANS] \\\\ \\hline 2 & 0.4 & [ANS] \\\\ \\hline 3 & 0.6 & [ANS] \\\\ \\hline 4 & 0.8 & [ANS] \\\\ \\hline 5 & 1.0 & [ANS] \\\\ \\hline \\end{array}$\n(b) Use separation of variables to find $y(t)$ exactly. $y(t)$=[ANS]\n(c) Compute the error in the approximations to $y(0.2),\\, y(0.6)$, and $y(1)$. $\\left|y(0.2)-y_1\\right|=$ [ANS]\n$\\left|y(0.6)-y_3\\right|=$ [ANS]\n$\\left|y(1)-y_{5}\\right|=$ [ANS]", "answer_v3": [ "1", "1.05886", "1.16985", "1.31885", "1.49001", "ln(2*t^2+2.71828)", "0.0290049", "0.0651184", "0.0614302" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Differential_equations_0275", "subject": "Differential_equations", "topic": "Numerical methods", "subtopic": "Euler", "level": "2", "keywords": [ "differential equation' 'euler" ], "problem_v1": "Use Euler's method with step size $0.3$ to estimate $y(1.5)$, where $y(x)$ is the solution of the initial-value problem y'=5x+y^2, \\ \\ \\ y(0)=1. $y(1.5)=$ [ANS].", "answer_v1": [ "62.204341027879" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use Euler's method with step size $0.1$ to estimate $y(0.5)$, where $y(x)$ is the solution of the initial-value problem y'=2x+y^2, \\ \\ \\ y(0)=0. $y(0.5)=$ [ANS].", "answer_v2": [ "0.201850107722276" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use Euler's method with step size $0.2$ to estimate $y(1)$, where $y(x)$ is the solution of the initial-value problem y'=3x+y^2, \\ \\ \\ y(0)=0. $y(1)=$ [ANS].", "answer_v3": [ "1.34148141535641" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0276", "subject": "Differential_equations", "topic": "Numerical methods", "subtopic": "Euler", "level": "2", "keywords": [ "calculus", "integral", "differential equations" ], "problem_v1": "Fill in the missing values in the table given if you know that $dy/dt=0.8 y$. Assume the rate of growth given by $dy/dt$ is approximately constant over each unit time interval and that the initial value of $y$ is $8$. $\\begin{array}{cccccc}\\hline t & 0 & 1 & 2 & 3 & 4 \\\\ \\hline y & 8 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "14.4", "25.92", "46.656", "83.9808" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Fill in the missing values in the table given if you know that $dy/dt=0.2 y$. Assume the rate of growth given by $dy/dt$ is approximately constant over each unit time interval and that the initial value of $y$ is $10$. $\\begin{array}{cccccc}\\hline t & 0 & 1 & 2 & 3 & 4 \\\\ \\hline y & 10 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "12", "14.4", "17.28", "20.736" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Fill in the missing values in the table given if you know that $dy/dt=0.4 y$. Assume the rate of growth given by $dy/dt$ is approximately constant over each unit time interval and that the initial value of $y$ is $8$. $\\begin{array}{cccccc}\\hline t & 0 & 1 & 2 & 3 & 4 \\\\ \\hline y & 8 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "11.2", "15.68", "21.952", "30.7328" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0277", "subject": "Differential_equations", "topic": "Numerical methods", "subtopic": "Euler", "level": "2", "keywords": [ "calculus", "integral", "differential equations", "euler", "numerical calculation" ], "problem_v1": "Consider the solution of the differential equation $y'=2 y$ passing through $y(0)=1$. A. Sketch the slope field for this differential equation, and sketch the solution passing through the point (0,1). B. Use Euler's method with step size $\\Delta x=0.2$ to estimate the solution at $x=0.2,0.4,\\ldots,1$, using these to fill in the following table. (Be sure not to round your answers at each step!)\n$\\begin{array}{ccccccc}\\hline x=& 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\\\ \\hline y\\approx & 1 & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nC. Plot your estimated solution on your slope field. Compare the solution and the slope field. Is the estimated solution an over or under estimate for the actual solution? [ANS] A. over B. under\nD. Check that $y=e^{2x}$ is a solution to $y'=2 y$ with $y(0)=1$.", "answer_v1": [ "1.4", "1.96", "2.744", "3.8416", "5.37824", "B" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [], [], [ "A", "B" ] ], "problem_v2": "Consider the solution of the differential equation $y'=-4 y$ passing through $y(0)=1.5$. A. Sketch the slope field for this differential equation, and sketch the solution passing through the point (0,1.5). B. Use Euler's method with step size $\\Delta x=0.2$ to estimate the solution at $x=0.2,0.4,\\ldots,1$, using these to fill in the following table. (Be sure not to round your answers at each step!)\n$\\begin{array}{ccccccc}\\hline x=& 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\\\ \\hline y\\approx & 1.5 & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nC. Plot your estimated solution on your slope field. Compare the solution and the slope field. Is the estimated solution an over or under estimate for the actual solution? [ANS] A. over B. under\nD. Check that $y=1.5 e^{-4x}$ is a solution to $y'=-4 y$ with $y(0)=1.5$.", "answer_v2": [ "0.3", "0.06", "0.012", "0.0024", "0.00048", "B" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [], [], [ "A", "B" ] ], "problem_v3": "Consider the solution of the differential equation $y'=-2 y$ passing through $y(0)=1$. A. Sketch the slope field for this differential equation, and sketch the solution passing through the point (0,1). B. Use Euler's method with step size $\\Delta x=0.2$ to estimate the solution at $x=0.2,0.4,\\ldots,1$, using these to fill in the following table. (Be sure not to round your answers at each step!)\n$\\begin{array}{ccccccc}\\hline x=& 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\\\ \\hline y\\approx & 1 & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nC. Plot your estimated solution on your slope field. Compare the solution and the slope field. Is the estimated solution an over or under estimate for the actual solution? [ANS] A. over B. under\nD. Check that $y=e^{-2x}$ is a solution to $y'=-2 y$ with $y(0)=1$.", "answer_v3": [ "0.6", "0.36", "0.216", "0.1296", "0.07776", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [], [], [ "A", "B" ] ] }, { "id": "Differential_equations_0278", "subject": "Differential_equations", "topic": "Numerical methods", "subtopic": "Euler", "level": "3", "keywords": [ "calculus", "integral", "differential equations", "euler", "numerical calculation" ], "problem_v1": "Use Euler's method to solve \\frac{dB}{dt} =0.07 B with initial value $B=1200$ when $t=0$. A. $\\Delta t=1$ and 1 step: $B(1) \\approx$ [ANS]\nB. $\\Delta t=0.5$ and 2 steps: $B(1) \\approx$ [ANS]\nC. $\\Delta t=0.25$ and 4 steps: $B(1) \\approx$ [ANS]\nD. Suppose $B$ is the balance in a bank account earning interest. Be sure that you can explain why the result of your calculation in part (a) is equivalent to compounding the interest once a year instead of continuously. Then interpret the result of your calculations in parts (b) and (c) in terms of compound interest.", "answer_v1": [ "(1 + 0.07)*1200", "1200*(1 + 0.07/2)^2", "1200*(1 + 0.07/4)^4" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Use Euler's method to solve \\frac{dB}{dt} =0.03 B with initial value $B=1500$ when $t=0$. A. $\\Delta t=1$ and 1 step: $B(1) \\approx$ [ANS]\nB. $\\Delta t=0.5$ and 2 steps: $B(1) \\approx$ [ANS]\nC. $\\Delta t=0.25$ and 4 steps: $B(1) \\approx$ [ANS]\nD. Suppose $B$ is the balance in a bank account earning interest. Be sure that you can explain why the result of your calculation in part (a) is equivalent to compounding the interest once a year instead of continuously. Then interpret the result of your calculations in parts (b) and (c) in terms of compound interest.", "answer_v2": [ "(1 + 0.03)*1500", "1500*(1 + 0.03/2)^2", "1500*(1 + 0.03/4)^4" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Use Euler's method to solve \\frac{dB}{dt} =0.04 B with initial value $B=1200$ when $t=0$. A. $\\Delta t=1$ and 1 step: $B(1) \\approx$ [ANS]\nB. $\\Delta t=0.5$ and 2 steps: $B(1) \\approx$ [ANS]\nC. $\\Delta t=0.25$ and 4 steps: $B(1) \\approx$ [ANS]\nD. Suppose $B$ is the balance in a bank account earning interest. Be sure that you can explain why the result of your calculation in part (a) is equivalent to compounding the interest once a year instead of continuously. Then interpret the result of your calculations in parts (b) and (c) in terms of compound interest.", "answer_v3": [ "(1 + 0.04)*1200", "1200*(1 + 0.04/2)^2", "1200*(1 + 0.04/4)^4" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0279", "subject": "Differential_equations", "topic": "Numerical methods", "subtopic": "Euler", "level": "4", "keywords": [ "calculus", "integral", "differential equations", "euler", "numerical calculation" ], "problem_v1": "Consider the differential equation \\frac{dy}{dx} =6x, with initial condition $y(0)=3$. A. Use Euler's method with two steps to estimate $y$ when $x=1$: $y(1) \\approx$ [ANS]\n(Be sure not to round your calculations at each step!) Now use four steps: $y(1) \\approx$ [ANS]\n(Be sure not to round your calculations at each step!) B. What is the solution to this differential equation (with the given initial condition)? $y=$ [ANS]\nC. What is the magnitude of the error in the two Euler approximations you found? Magnitude of error in Euler with 2 steps=[ANS]\nMagnitude of error in Euler with 4 steps=[ANS]\nD. By what factor should the error in these approximations change (that is, the error with two steps should be what number times the error with four)? factor=[ANS]\n(How close to this is the result you obtained above?) (How close to this is the result you obtained above?)", "answer_v1": [ "3 + 0.25*6", "3 + 0.25*6*(0.25 + 0.5 + 0.75)", "3 + (6/2)*x^2", "1.5", "0.75", "2" ], "answer_type_v1": [ "NV", "NV", "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Consider the differential equation \\frac{dy}{dx} =2x, with initial condition $y(0)=5$. A. Use Euler's method with two steps to estimate $y$ when $x=1$: $y(1) \\approx$ [ANS]\n(Be sure not to round your calculations at each step!) Now use four steps: $y(1) \\approx$ [ANS]\n(Be sure not to round your calculations at each step!) B. What is the solution to this differential equation (with the given initial condition)? $y=$ [ANS]\nC. What is the magnitude of the error in the two Euler approximations you found? Magnitude of error in Euler with 2 steps=[ANS]\nMagnitude of error in Euler with 4 steps=[ANS]\nD. By what factor should the error in these approximations change (that is, the error with two steps should be what number times the error with four)? factor=[ANS]\n(How close to this is the result you obtained above?) (How close to this is the result you obtained above?)", "answer_v2": [ "5 + 0.25*2", "5 + 0.25*2*(0.25 + 0.5 + 0.75)", "5 + (2/2)*x^2", "0.5", "0.25", "2" ], "answer_type_v2": [ "NV", "NV", "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Consider the differential equation \\frac{dy}{dx} =3x, with initial condition $y(0)=4$. A. Use Euler's method with two steps to estimate $y$ when $x=1$: $y(1) \\approx$ [ANS]\n(Be sure not to round your calculations at each step!) Now use four steps: $y(1) \\approx$ [ANS]\n(Be sure not to round your calculations at each step!) B. What is the solution to this differential equation (with the given initial condition)? $y=$ [ANS]\nC. What is the magnitude of the error in the two Euler approximations you found? Magnitude of error in Euler with 2 steps=[ANS]\nMagnitude of error in Euler with 4 steps=[ANS]\nD. By what factor should the error in these approximations change (that is, the error with two steps should be what number times the error with four)? factor=[ANS]\n(How close to this is the result you obtained above?) (How close to this is the result you obtained above?)", "answer_v3": [ "4 + 0.25*3", "4 + 0.25*3*(0.25 + 0.5 + 0.75)", "4 + (3/2)*x^2", "0.75", "0.375", "2" ], "answer_type_v3": [ "NV", "NV", "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Differential_equations_0280", "subject": "Differential_equations", "topic": "Numerical methods", "subtopic": "Euler", "level": "2", "keywords": [ "derivative" ], "problem_v1": "It can be shown that the solution to the initial value problem y'=e^{-\\left( \\frac{x}{5} \\right)^{2}}, y(3)=0 is y(x)= \\int_{3}^{x} e^{-\\left( \\frac{t}{5} \\right)^{2}}\\; dt Use Euler's Method with $\\Delta x=0.1$ to approximate $y(5)= \\int_{3}^{5} e^{-\\left( \\frac{t}{5} \\right)^{2}}\\; dt$ to six decimal places (do not round intermediate results). $y(5)\\approx$ [ANS]", "answer_v1": [ "1.07486" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "It can be shown that the solution to the initial value problem y'=e^{-\\left( \\frac{x}{5} \\right)^{2}}, y(0)=0 is y(x)= \\int_{0}^{x} e^{-\\left( \\frac{t}{5} \\right)^{2}}\\; dt Use Euler's Method with $\\Delta x=0.15$ to approximate $y(3)= \\int_{0}^{3} e^{-\\left( \\frac{t}{5} \\right)^{2}}\\; dt$ to six decimal places (do not round intermediate results). $y(3)\\approx$ [ANS]", "answer_v2": [ "2.69813" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "It can be shown that the solution to the initial value problem y'=e^{-\\left( \\frac{x}{5} \\right)^{2}}, y(1)=0 is y(x)= \\int_{1}^{x} e^{-\\left( \\frac{t}{5} \\right)^{2}}\\; dt Use Euler's Method with $\\Delta x=0.1$ to approximate $y(3)= \\int_{1}^{3} e^{-\\left( \\frac{t}{5} \\right)^{2}}\\; dt$ to six decimal places (do not round intermediate results). $y(3)\\approx$ [ANS]", "answer_v3": [ "1.70202" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0281", "subject": "Differential_equations", "topic": "Numerical methods", "subtopic": "Euler", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Suppose that we use the Improved Euler's method to approximate the solution to the differential equation \\frac{dy}{dx} =x-2 y;\\qquad y(0.4)=6. Let $f(x,y)=x-2 y.$ We let $x_0=0.4$ and $y_0=6$ and pick a step size $h=0.25.$ The improved Euler method is the following algorithm. From $(x_n, y_n),$ our approximation to the solution of the differential equation at the $n$-th stage, we find the next stage by computing the $x$-step $x_{n+1}=x_n+h,$ and then $k_1,$ the slope at $(x_n,y_n).$ The predicted new value of the solution is $z_{n+1}=y_n+h\\cdot k_1.$ Then we find the slope at the predicted new point $k_2=f(x_{n+1}, z_{n+1})$ and get the corrected point by averaging slopes y_{n+1}=y_n+\\frac h2 (k_1+k_2). Complete the following table:\n$\\begin{array}{cccccc}\\hline n & x_n & y_n & k_1 & z_{n+1} & k_2 \\\\ \\hline 0 & 0.4 & 6 &-11.6 & 3.1 &-5.55 \\\\ \\hline 1 & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 2 & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 3 & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 4 & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nThe exact solution can also be found for the linear equation. Write the answer as a function of $x$. $y(x)=$ [ANS]\nThus the actual value of the function at the point $x=1.4$ is $y(1.4)=$ [ANS].", "answer_v1": [ "0.65", "3.85625", "-7.0625", "2.090625", "-3.28125", "0.9", "2.56328125", "-4.2265625", "1.506640625", "-1.86328125", "1.15", "1.80205078125", "-2.4541015625", "1.188525390625", "-0.97705078125", "1.4", "1.37315673828125", "x/2-0.25 + 13.4645226173794*exp(-2*x)", "1.4/2-0.25 + 13.4645226173794*exp(-2*1.4)" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Suppose that we use the Improved Euler's method to approximate the solution to the differential equation \\frac{dy}{dx} =x-0.5 y;\\qquad y(0)=9. Let $f(x,y)=x-0.5 y.$ We let $x_0=0$ and $y_0=9$ and pick a step size $h=0.25.$ The improved Euler method is the following algorithm. From $(x_n, y_n),$ our approximation to the solution of the differential equation at the $n$-th stage, we find the next stage by computing the $x$-step $x_{n+1}=x_n+h,$ and then $k_1,$ the slope at $(x_n,y_n).$ The predicted new value of the solution is $z_{n+1}=y_n+h\\cdot k_1.$ Then we find the slope at the predicted new point $k_2=f(x_{n+1}, z_{n+1})$ and get the corrected point by averaging slopes y_{n+1}=y_n+\\frac h2 (k_1+k_2). Complete the following table:\n$\\begin{array}{cccccc}\\hline n & x_n & y_n & k_1 & z_{n+1} & k_2 \\\\ \\hline 0 & 0 & 9 &-4.5 & 7.875 &-3.6875 \\\\ \\hline 1 & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 2 & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 3 & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 4 & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nThe exact solution can also be found for the linear equation. Write the answer as a function of $x$. $y(x)=$ [ANS]\nThus the actual value of the function at the point $x=1$ is $y(1)=$ [ANS].", "answer_v2": [ "0.25", "7.9765625", "-3.73828125", "7.0419921875", "-3.02099609375", "0.5", "7.13165283203125", "-3.06582641601562", "6.36519622802734", "-2.43259811401367", "0.75", "6.44434976577759", "-2.47217488288879", "5.82630604505539", "-1.91315302252769", "1", "5.89618377760053", "x/0.5-4 + 13*exp(-0.5*x)", "1/0.5-4 + 13*exp(-0.5*1)" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Suppose that we use the Improved Euler's method to approximate the solution to the differential equation \\frac{dy}{dx} =x-1 y;\\qquad y(0.1)=6. Let $f(x,y)=x-1 y.$ We let $x_0=0.1$ and $y_0=6$ and pick a step size $h=0.25.$ The improved Euler method is the following algorithm. From $(x_n, y_n),$ our approximation to the solution of the differential equation at the $n$-th stage, we find the next stage by computing the $x$-step $x_{n+1}=x_n+h,$ and then $k_1,$ the slope at $(x_n,y_n).$ The predicted new value of the solution is $z_{n+1}=y_n+h\\cdot k_1.$ Then we find the slope at the predicted new point $k_2=f(x_{n+1}, z_{n+1})$ and get the corrected point by averaging slopes y_{n+1}=y_n+\\frac h2 (k_1+k_2). Complete the following table:\n$\\begin{array}{cccccc}\\hline n & x_n & y_n & k_1 & z_{n+1} & k_2 \\\\ \\hline 0 & 0.1 & 6 &-5.9 & 4.525 &-4.175 \\\\ \\hline 1 & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 2 & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 3 & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 4 & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nThe exact solution can also be found for the linear equation. Write the answer as a function of $x$. $y(x)=$ [ANS]\nThus the actual value of the function at the point $x=1.1$ is $y(1.1)=$ [ANS].", "answer_v3": [ "0.35", "4.740625", "-4.390625", "3.64296875", "-3.04296875", "0.6", "3.81142578125", "-3.21142578125", "3.0085693359375", "-2.1585693359375", "0.85", "3.14017639160156", "-2.29017639160156", "2.56763229370117", "-1.46763229370117", "1.1", "2.67045030593872", "x/1-1 + 7.62567933472197*exp(-1*x)", "1.1/1-1 + 7.62567933472197*exp(-1*1.1)" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Differential_equations_0282", "subject": "Differential_equations", "topic": "Series solutions", "subtopic": "Ordinary point", "level": "4", "keywords": [ "power series solution" ], "problem_v1": "Find two linearly independent solutions of $y''+8xy=0$ of the form\n$y_1=1+a_3x^3+a_6x^6+\\cdots$\n$y_2=x+b_4x^4+b_7x^7+\\cdots$\nEnter the first few coefficients:\n$a_3=$ [ANS]\n$a_6=$ [ANS]\n$b_4=$ [ANS]\n$b_7=$ [ANS]", "answer_v1": [ "-8/6", "8^2/180", "-8/12", "8^2/504" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find two linearly independent solutions of $y''+1xy=0$ of the form\n$y_1=1+a_3x^3+a_6x^6+\\cdots$\n$y_2=x+b_4x^4+b_7x^7+\\cdots$\nEnter the first few coefficients:\n$a_3=$ [ANS]\n$a_6=$ [ANS]\n$b_4=$ [ANS]\n$b_7=$ [ANS]", "answer_v2": [ "-1/6", "1^2/180", "-1/12", "1^2/504" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find two linearly independent solutions of $y''+4xy=0$ of the form\n$y_1=1+a_3x^3+a_6x^6+\\cdots$\n$y_2=x+b_4x^4+b_7x^7+\\cdots$\nEnter the first few coefficients:\n$a_3=$ [ANS]\n$a_6=$ [ANS]\n$b_4=$ [ANS]\n$b_7=$ [ANS]", "answer_v3": [ "-4/6", "4^2/180", "-4/12", "4^2/504" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0284", "subject": "Differential_equations", "topic": "Series solutions", "subtopic": "Ordinary point", "level": "2", "keywords": [ "differential equation", "second order", "homogeneous", "series solution" ], "problem_v1": "If y=\\sum_{n=0}^\\infty c_nx^n is a solution of the differential equation y''+(2x+1)y'+1 y=0 \\;, then its coefficients $c_n$ are related by the equation\n$c_{n+2}=$ [ANS] $c_{n+1}$+[ANS] $c_n$.", "answer_v1": [ "-1/(n+2)", "-(2*n+1)/[(n+1)*(n+2)]" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "If y=\\sum_{n=0}^\\infty c_nx^n is a solution of the differential equation y''+(-4x+3)y'-2 y=0 \\;, then its coefficients $c_n$ are related by the equation\n$c_{n+2}=$ [ANS] $c_{n+1}$+[ANS] $c_n$.", "answer_v2": [ "-3/(n+2)", "-(-4*n+-2)/[(n+1)*(n+2)]" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "If y=\\sum_{n=0}^\\infty c_nx^n is a solution of the differential equation y''+(-2x+1)y'-2 y=0 \\;, then its coefficients $c_n$ are related by the equation\n$c_{n+2}=$ [ANS] $c_{n+1}$+[ANS] $c_n$.", "answer_v3": [ "-1/(n+2)", "-(-2*n+-2)/[(n+1)*(n+2)]" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0285", "subject": "Differential_equations", "topic": "Series solutions", "subtopic": "Ordinary point", "level": "2", "keywords": [ "differential equation", "first order", "linear", "nonhomogeneous", "series solution" ], "problem_v1": "Assume that $y$ is the solution of the initial-value problem y'+2 y=\\begin{cases} \\frac{4\\sin x}{x} & x \\ne 0 \\\\ 4 & x=0 \\end{cases}\\;, \\qquad y(0)=1 \\;. If $y$ is written as a power series y=\\sum_{n=0}^\\infty c_nx^n\\;, then $y=$ [ANS]+[ANS] $x$+[ANS] $x^2$+[ANS] $x^3$+[ANS] $x^4+\\cdots$.\nNote: You do not have to find a general expression for $c_n$. Just find the coefficients one by one.", "answer_v1": [ "1", "2", "2*(2-4)/2", "-4/18+2*2*(4-2)/6", "-1.11111*2/4" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Assume that $y$ is the solution of the initial-value problem y'-4 y=\\begin{cases} \\frac{5\\sin x}{x} & x \\ne 0 \\\\ 5 & x=0 \\end{cases}\\;, \\qquad y(0)=1 \\;. If $y$ is written as a power series y=\\sum_{n=0}^\\infty c_nx^n\\;, then $y=$ [ANS]+[ANS] $x$+[ANS] $x^2$+[ANS] $x^3$+[ANS] $x^4+\\cdots$.\nNote: You do not have to find a general expression for $c_n$. Just find the coefficients one by one.", "answer_v2": [ "1", "9", "-4*(-4-5)/2", "-5/18+-4*-4*(5--4)/6", "-23.7222*-4/4" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Assume that $y$ is the solution of the initial-value problem y'-2 y=\\begin{cases} \\frac{4\\sin x}{x} & x \\ne 0 \\\\ 4 & x=0 \\end{cases}\\;, \\qquad y(0)=1 \\;. If $y$ is written as a power series y=\\sum_{n=0}^\\infty c_nx^n\\;, then $y=$ [ANS]+[ANS] $x$+[ANS] $x^2$+[ANS] $x^3$+[ANS] $x^4+\\cdots$.\nNote: You do not have to find a general expression for $c_n$. Just find the coefficients one by one.", "answer_v3": [ "1", "6", "-2*(-2-4)/2", "-4/18+-2*-2*(4--2)/6", "-3.77778*-2/4" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Differential_equations_0286", "subject": "Differential_equations", "topic": "Series solutions", "subtopic": "Singular point", "level": "3", "keywords": [ "regular singular point" ], "problem_v1": "Find the solution of $4x^2y''+3x^2y'+y=0, \\, x>0$ of the form\n$y_1=x^r(1+c_1x+c_2x^2+c_3x^3+\\cdots)$\nEnter\n$r=$ [ANS]\n$c_1=$ [ANS]\n$c_2=$ [ANS]\n$c_3=$ [ANS]", "answer_v1": [ "1/2", "-3/8", "3*(3)^2/256", "-5*(3)^3/6144" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find the solution of $4x^2y''-5x^2y'+y=0, \\, x>0$ of the form\n$y_1=x^r(1+c_1x+c_2x^2+c_3x^3+\\cdots)$\nEnter\n$r=$ [ANS]\n$c_1=$ [ANS]\n$c_2=$ [ANS]\n$c_3=$ [ANS]", "answer_v2": [ "1/2", "--5/8", "3*(-5)^2/256", "-5*(-5)^3/6144" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find the solution of $4x^2y''-2x^2y'+y=0, \\, x>0$ of the form\n$y_1=x^r(1+c_1x+c_2x^2+c_3x^3+\\cdots)$\nEnter\n$r=$ [ANS]\n$c_1=$ [ANS]\n$c_2=$ [ANS]\n$c_3=$ [ANS]", "answer_v3": [ "1/2", "--2/8", "3*(-2)^2/256", "-5*(-2)^3/6144" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0288", "subject": "Differential_equations", "topic": "Series solutions", "subtopic": "Singular point", "level": "3", "keywords": [ "differential equation", "classify", "singular", "regular", "Frobenius" ], "problem_v1": "Write the differential equation $ \\frac{x^{2}-64}{x} y''+5y'+y=0$ in standard form:\n$\\begin{array}{ccccccccccc}\\hline [ANS] & & y^{\\prime\\prime}+& & \\frac{[ANS]}{[ANS]} & & y^{\\prime}+& & \\frac{[ANS]}{[ANS]} & & y=0 \\\\ \\hline \\end{array}$\nList the singular points of the differential equation: $x=$ [ANS]", "answer_v1": [ "1", "x^2-8^2", "5*x", "x^2-8^2", "x", "(8, -8)" ], "answer_type_v1": [ "NV", "EX", "EX", "EX", "EX", "UOL" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Write the differential equation $ \\frac{x^{2}-4}{x} y''+7y'+y=0$ in standard form:\n$\\begin{array}{ccccccccccc}\\hline [ANS] & & y^{\\prime\\prime}+& & \\frac{[ANS]}{[ANS]} & & y^{\\prime}+& & \\frac{[ANS]}{[ANS]} & & y=0 \\\\ \\hline \\end{array}$\nList the singular points of the differential equation: $x=$ [ANS]", "answer_v2": [ "1", "x^2-2^2", "7*x", "x^2-2^2", "x", "(2, -2)" ], "answer_type_v2": [ "NV", "EX", "EX", "EX", "EX", "UOL" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Write the differential equation $ \\frac{x^{2}-16}{x} y''+6y'+y=0$ in standard form:\n$\\begin{array}{ccccccccccc}\\hline [ANS] & & y^{\\prime\\prime}+& & \\frac{[ANS]}{[ANS]} & & y^{\\prime}+& & \\frac{[ANS]}{[ANS]} & & y=0 \\\\ \\hline \\end{array}$\nList the singular points of the differential equation: $x=$ [ANS]", "answer_v3": [ "1", "x^2-4^2", "6*x", "x^2-4^2", "x", "(4, -4)" ], "answer_type_v3": [ "NV", "EX", "EX", "EX", "EX", "UOL" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Differential_equations_0289", "subject": "Differential_equations", "topic": "Series solutions", "subtopic": "Bessel functions", "level": "3", "keywords": [ "bessel", "series", "singular point" ], "problem_v1": "The general solution of the differential equation $x^2y^{\\prime\\prime}+xy^\\prime+(x^2- \\frac{1}{64} )y=0$ is? Check all that apply. [ANS] A. $2 J_{(1/8)}(x)+2 Y_{(1/8)}(x)$ B. $c_1J_{8}(x)+c_2J_{-8}(x)$ C. $c_1J_{(1/8)}(x)+c_2J_{-(1/8)}(x)$ D. $2 J_{(1/8)}(x)+2 J_{-(1/8)}(x)$ E. $c_1J_{(1/8)}(x)+c_2Y_{(1/8)}(x)$ F. $2 J_{8}(x)+2 J_{-8}(x)$", "answer_v1": [ "CE" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "The general solution of the differential equation $x^2y^{\\prime\\prime}+xy^\\prime+(x^2- \\frac{1}{4} )y=0$ is? Check all that apply. [ANS] A. $8 J_{(1/2)}(x)-7 Y_{(1/2)}(x)$ B. $c_1J_{(1/2)}(x)+c_2J_{-(1/2)}(x)$ C. $c_1J_{(1/2)}(x)+c_2Y_{(1/2)}(x)$ D. $8 J_{(1/2)}(x)-7 J_{-(1/2)}(x)$ E. $c_1J_{2}(x)+c_2J_{-2}(x)$ F. $8 J_{2}(x)-7 J_{-2}(x)$", "answer_v2": [ "BC" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "The general solution of the differential equation $x^2y^{\\prime\\prime}+xy^\\prime+(x^2- \\frac{1}{16} )y=0$ is? Check all that apply. [ANS] A. $c_1J_{4}(x)+c_2J_{-4}(x)$ B. $c_1J_{(1/4)}(x)+c_2Y_{(1/4)}(x)$ C. $2 J_{(1/4)}(x)-4 Y_{(1/4)}(x)$ D. $c_1J_{(1/4)}(x)+c_2J_{-(1/4)}(x)$ E. $2 J_{(1/4)}(x)-4 J_{-(1/4)}(x)$ F. $2 J_{4}(x)-4 J_{-4}(x)$", "answer_v3": [ "BD" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Differential_equations_0290", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Classification", "level": "2", "keywords": [ "PDE", "partial", "differential", "equation", "classify" ], "problem_v1": "Classify the equations in terms of order, linearity, homogeneity, and for second order equations classify as elliptic, hyperbolic, or parabolic.\n[ANS] 1. $u_{xx}=u_{tt}+u_x+u+xt$ [ANS] 2. $u_{t}+u_{xx}=u$ [ANS] 3. $u_{xx}+u_{yy}=0$ [ANS] 4. $u_x+3u_t=u$ [ANS] 5. $(u_{xy})^2=u$ [ANS] 6. $u_{xx}+u^2-u_{yy}=0$ [ANS] 7. $u^2u_t+u_x+u_y=xy$", "answer_v1": [ "SECOND ORDER, LINEAR, NONHOMOGENEOUS, HYPERBOLIC", "SECOND ORDER, LINEAR, HOMOGENEOUS, PARABOLIC", "SECOND ORDER, LINEAR, HOMOGENEOUS, ELLIPTIC", "First order, linear, homogeneous", "Second order, nonlinear", "Second order, nonlinear", "First order, nonlinear" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ] ], "problem_v2": "Classify the equations in terms of order, linearity, homogeneity, and for second order equations classify as elliptic, hyperbolic, or parabolic.\n[ANS] 1. $(u_{xy})^2=u$ [ANS] 2. $u_{t}+u_{xx}=u$ [ANS] 3. $u_{xx}+u^2-u_{yy}=0$ [ANS] 4. $u_{xx}=u_{tt}+u_x+u+xt$ [ANS] 5. $u_{xx}+u_{yy}=0$ [ANS] 6. $u^2u_t+u_x+u_y=xy$ [ANS] 7. $u_x+3u_t=u$", "answer_v2": [ "SECOND ORDER, NONLINEAR", "SECOND ORDER, LINEAR, HOMOGENEOUS, PARABOLIC", "SECOND ORDER, NONLINEAR", "Second order, linear, nonhomogeneous, hyperbolic", "Second order, linear, homogeneous, elliptic", "First order, nonlinear", "First order, linear, homogeneous" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ] ], "problem_v3": "Classify the equations in terms of order, linearity, homogeneity, and for second order equations classify as elliptic, hyperbolic, or parabolic.\n[ANS] 1. $u_{xx}+u_{yy}=0$ [ANS] 2. $u_{xx}+u^2-u_{yy}=0$ [ANS] 3. $u_{xx}=u_{tt}+u_x+u+xt$ [ANS] 4. $u_x+3u_t=u$ [ANS] 5. $(u_{xy})^2=u$ [ANS] 6. $u_{t}+u_{xx}=u$ [ANS] 7. $u^2u_t+u_x+u_y=xy$", "answer_v3": [ "SECOND ORDER, LINEAR, HOMOGENEOUS, ELLIPTIC", "SECOND ORDER, NONLINEAR", "SECOND ORDER, LINEAR, NONHOMOGENEOUS, HYPERBOLIC", "First order, linear, homogeneous", "Second order, nonlinear", "Second order, linear, homogeneous, parabolic", "First order, nonlinear" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ], [ "Second order", "linear", "homogeneous", "elliptic", "linear", "homogeneous", "elliptic", "Second order", "linear", "homogeneous", "hyperbolic", "linear", "homogeneous", "hyperbolic", "Second order", "linear", "homogeneous", "parabolic", "linear", "homogeneous", "parabolic", "Second order", "linear", "nonhomogeneous", "elliptic", "linear", "nonhomogeneous", "elliptic", "Second order", "linear", "nonhomogeneous", "hyperbolic", "linear", "nonhomogeneous", "hyperbolic", "Second order", "linear", "nonhomogeneous", "parabolic", "linear", "nonhomogeneous", "parabolic", "Second order", "nonlinear", "nonlinear", "First order", "linear", "homogeneous", "linear", "homogeneous", "First order", "linear", "nonhomogeneous", "linear", "nonhomogeneous", "First order", "nonlinear", "nonlinear" ] ] }, { "id": "Differential_equations_0291", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Classification", "level": "2", "keywords": [ "partial", "differential", "equation", "classify", "separable" ], "problem_v1": "For the partial differential equation $2 \\frac{\\partial^2 u}{\\partial x^2} =-5 \\frac{\\partial^2 u}{\\partial t^2} $ the discriminant is [ANS]\nThe PDE is (check all that apply) [ANS] A. elliptic B. inseparable C. hyperbolic D. separable E. parabolic", "answer_v1": [ "0^2-4*5*2", "AD" ], "answer_type_v1": [ "NV", "MCM" ], "options_v1": [ [], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "For the partial differential equation $8 \\frac{\\partial^2 u}{\\partial x^2} =8 \\frac{\\partial^2 u}{\\partial t^2} $ the discriminant is [ANS]\nThe PDE is (check all that apply) [ANS] A. separable B. hyperbolic C. inseparable D. parabolic E. elliptic", "answer_v2": [ "0^2-4*-8*8", "AB" ], "answer_type_v2": [ "NV", "MCM" ], "options_v2": [ [], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "For the partial differential equation $2 \\frac{\\partial^2 u}{\\partial x^2} =4 \\frac{\\partial^2 u}{\\partial t^2} $ the discriminant is [ANS]\nThe PDE is (check all that apply) [ANS] A. separable B. hyperbolic C. parabolic D. inseparable E. elliptic", "answer_v3": [ "0^2-4*-4*2", "AB" ], "answer_type_v3": [ "NV", "MCM" ], "options_v3": [ [], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Differential_equations_0292", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Verification of solutions", "level": "3", "keywords": [ "PDE", "partial", "differential", "equation", "solution" ], "problem_v1": "Select all solutions of $u_t=6 u_{x}$. There may be more than one correct answer. [ANS] A. $u=6 e^{x}$ B. $u=x+6 t$ C. $u=0$ D. $u=6x+t$ E. $u=e^{x+6 t}$ F. $u=6x-t$ G. $u=e^{6x+t}$ H. None of the above", "answer_v1": [ "BCE" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v2": "Select all solutions of $u_t=2 u_{x}$. There may be more than one correct answer. [ANS] A. $u=e^{2x+t}$ B. $u=2 e^{x}$ C. $u=0$ D. $u=x+2 t$ E. $u=2x+t$ F. $u=e^{x+2 t}$ G. $u=2x-t$ H. None of the above", "answer_v2": [ "CDF" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G", "H" ] ], "problem_v3": "Select all solutions of $u_t=3 u_{x}$. There may be more than one correct answer. [ANS] A. $u=0$ B. $u=3x-t$ C. $u=3x+t$ D. $u=e^{x+3 t}$ E. $u=e^{3x+t}$ F. $u=3 e^{x}$ G. $u=x+3 t$ H. None of the above", "answer_v3": [ "ADG" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G", "H" ] ] }, { "id": "Differential_equations_0293", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Heat equation", "level": "4", "keywords": [ "PDE", "partial", "differential", "equation", "heat equation", "dimensionless" ], "problem_v1": "We want to solve the problem PDE: $\\quad u_{t}=64 u_{xx}, \\qquad 0 < x < 6, \\quad t > 0$ BC: $\\quad u(0,t)=7 \\quad u(6,t)=4$ IC: $\\quad u(x,0)=0$ And we have the solution of the problem PDE: $\\quad v_{\\tau}=v_{\\xi\\xi}, \\qquad 0 < \\xi < 1, \\quad \\tau > 0$ BC: $\\quad v(0,\\tau)=0 \\quad v(1,\\tau)=1$ IC: $\\quad v(\\xi,0)=0$\nWrite $u(x,t)$ in terms of $v(\\xi,\\tau)$ $u(x,t)={}$ [ANS]", "answer_v1": [ "(4-7)*v(x/6,(8/6)^2*t)+7" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "We want to solve the problem PDE: $\\quad u_{t}=4 u_{xx}, \\qquad 0 < x < 9, \\quad t > 0$ BC: $\\quad u(0,t)=3 \\quad u(9,t)=4$ IC: $\\quad u(x,0)=0$ And we have the solution of the problem PDE: $\\quad v_{\\tau}=v_{\\xi\\xi}, \\qquad 0 < \\xi < 1, \\quad \\tau > 0$ BC: $\\quad v(0,\\tau)=0 \\quad v(1,\\tau)=1$ IC: $\\quad v(\\xi,0)=0$\nWrite $u(x,t)$ in terms of $v(\\xi,\\tau)$ $u(x,t)={}$ [ANS]", "answer_v2": [ "(4-3)*v(x/9,(2/9)^2*t)+3" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "We want to solve the problem PDE: $\\quad u_{t}=16 u_{xx}, \\qquad 0 < x < 6, \\quad t > 0$ BC: $\\quad u(0,t)=3 \\quad u(6,t)=8$ IC: $\\quad u(x,0)=0$ And we have the solution of the problem PDE: $\\quad v_{\\tau}=v_{\\xi\\xi}, \\qquad 0 < \\xi < 1, \\quad \\tau > 0$ BC: $\\quad v(0,\\tau)=0 \\quad v(1,\\tau)=1$ IC: $\\quad v(\\xi,0)=0$\nWrite $u(x,t)$ in terms of $v(\\xi,\\tau)$ $u(x,t)={}$ [ANS]", "answer_v3": [ "(8-3)*v(x/6,(4/6)^2*t)+3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0294", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Heat equation", "level": "4", "keywords": [ "PDE", "partial", "differential", "equation", "heat equation" ], "problem_v1": "Consider the equation $u_t=u_{xx}-8 u$, $0 < x < 1$, $t > 0$, with boundary condition $u(0,t)=0, u(1,t)=6 \\sinh(\\sqrt{8})$, and initial condition $u(x,0)=6\\sinh\\!\\left(\\sqrt{8}x\\right)+7\\sin\\!\\left(3\\pi x\\right)$.\nWe wish to solve it. Do this in two steps: first make the boundary conditions homogeneous, then get rid of the heat loss term. It may be useful to start by solving for the steady state.\n$u_{\\text{steady state}}(x)={}$ [ANS]\nThen the final solution is $u(x,t)={}$ [ANS]", "answer_v1": [ "6*sinh(sqrt(8)*x)", "6*sinh(sqrt(8)*x)+7*e^(-[(3*pi)^2+8]*t)*sin(3*pi*x)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Consider the equation $u_t=u_{xx}-2 u$, $0 < x < 1$, $t > 0$, with boundary condition $u(0,t)=0, u(1,t)=9 \\sinh(\\sqrt{2})$, and initial condition $u(x,0)=9\\sinh\\!\\left(\\sqrt{2}x\\right)+4\\sin\\!\\left(2\\pi x\\right)$.\nWe wish to solve it. Do this in two steps: first make the boundary conditions homogeneous, then get rid of the heat loss term. It may be useful to start by solving for the steady state.\n$u_{\\text{steady state}}(x)={}$ [ANS]\nThen the final solution is $u(x,t)={}$ [ANS]", "answer_v2": [ "9*sinh(sqrt(2)*x)", "9*sinh(sqrt(2)*x)+4*e^(-[(2*pi)^2+2]*t)*sin(2*pi*x)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Consider the equation $u_t=u_{xx}-4 u$, $0 < x < 1$, $t > 0$, with boundary condition $u(0,t)=0, u(1,t)=6 \\sinh(\\sqrt{4})$, and initial condition $u(x,0)=6\\sinh\\!\\left(\\sqrt{4}x\\right)+6\\sin\\!\\left(2\\pi x\\right)$.\nWe wish to solve it. Do this in two steps: first make the boundary conditions homogeneous, then get rid of the heat loss term. It may be useful to start by solving for the steady state.\n$u_{\\text{steady state}}(x)={}$ [ANS]\nThen the final solution is $u(x,t)={}$ [ANS]", "answer_v3": [ "6*sinh(sqrt(4)*x)", "6*sinh(sqrt(4)*x)+6*e^(-[(2*pi)^2+4]*t)*sin(2*pi*x)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0295", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Heat equation", "level": "4", "keywords": [ "PDE", "partial", "differential", "equation", "heat equation" ], "problem_v1": "The following scenario describes the temperature $u$ of a rod at position $x$ and time $t$. Consider the equation $u_t=u_{xx}$, $0 < x < 1$, $t > 0$, with boundary conditions $u(0,t)=0, u(1,t)=0$. Suppose $u(x,0)=8 \\sin(3 \\pi x)$\nWhat is the maximum temperature in the rod at any particular time. That is, $M(t)={}$ [ANS]\nwhere $M(t)$ is the maximum temperature at time $t$. Use your intuition.", "answer_v1": [ "8*e^[-(3*pi)^2*t]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The following scenario describes the temperature $u$ of a rod at position $x$ and time $t$. Consider the equation $u_t=u_{xx}$, $0 < x < 1$, $t > 0$, with boundary conditions $u(0,t)=0, u(1,t)=0$. Suppose $u(x,0)=2 \\sin(4 \\pi x)$\nWhat is the maximum temperature in the rod at any particular time. That is, $M(t)={}$ [ANS]\nwhere $M(t)$ is the maximum temperature at time $t$. Use your intuition.", "answer_v2": [ "2*e^[-(4*pi)^2*t]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The following scenario describes the temperature $u$ of a rod at position $x$ and time $t$. Consider the equation $u_t=u_{xx}$, $0 < x < 1$, $t > 0$, with boundary conditions $u(0,t)=0, u(1,t)=0$. Suppose $u(x,0)=4 \\sin(3 \\pi x)$\nWhat is the maximum temperature in the rod at any particular time. That is, $M(t)={}$ [ANS]\nwhere $M(t)$ is the maximum temperature at time $t$. Use your intuition.", "answer_v3": [ "4*e^[-(3*pi)^2*t]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0296", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Heat equation", "level": "3", "keywords": [ "PDE", "partial", "differential", "equation", "heat equation" ], "problem_v1": "Consider the equation $u_t=u_{xx}+10$, $0 < x < 1$, $t > 0$, that is, there is uniform heat source. Suppose further that ends are kept as $u(0,t)=0, u(1,t)=2$. What is the steady state solution $U(x)$, if it exists? That is, a solution after a very long time. Hint: a steady-state solution does not depend on time. $U(x)=$ [ANS]", "answer_v1": [ "7*x-5*x^2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Consider the equation $u_t=u_{xx}+4$, $0 < x < 1$, $t > 0$, that is, there is uniform heat source. Suppose further that ends are kept as $u(0,t)=0, u(1,t)=8$. What is the steady state solution $U(x)$, if it exists? That is, a solution after a very long time. Hint: a steady-state solution does not depend on time. $U(x)=$ [ANS]", "answer_v2": [ "10*x-2*x^2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Consider the equation $u_t=u_{xx}+6$, $0 < x < 1$, $t > 0$, that is, there is uniform heat source. Suppose further that ends are kept as $u(0,t)=0, u(1,t)=2$. What is the steady state solution $U(x)$, if it exists? That is, a solution after a very long time. Hint: a steady-state solution does not depend on time. $U(x)=$ [ANS]", "answer_v3": [ "5*x-3*x^2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0297", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Heat equation", "level": "2", "keywords": [ "PDE", "partial", "differential", "equation", "heat equation" ], "problem_v1": "Consider the equation $u_t=u_{xx}$, $0 < x < 1$, $t > 0$, with boundary conditions $u(0,t)=0, u(1,t)=0$. Suppose $ u(x,0)=\\sum_{n=1}^\\infty \\frac{5}{n^{7} } \\sin(n \\pi x).$\nThen the solution is\n$ u(x,t)=\\sum_{n=1}^\\infty$ [ANS] $e^{-(n\\pi)^2t} \\sin(n\\pi x)$", "answer_v1": [ "5/(n^7)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Consider the equation $u_t=u_{xx}$, $0 < x < 1$, $t > 0$, with boundary conditions $u(0,t)=0, u(1,t)=0$. Suppose $ u(x,0)=\\sum_{n=1}^\\infty \\frac{-8}{n^{10} } \\sin(n \\pi x).$\nThen the solution is\n$ u(x,t)=\\sum_{n=1}^\\infty$ [ANS] $e^{-(n\\pi)^2t} \\sin(n\\pi x)$", "answer_v2": [ "-[8/(n^10)]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Consider the equation $u_t=u_{xx}$, $0 < x < 1$, $t > 0$, with boundary conditions $u(0,t)=0, u(1,t)=0$. Suppose $ u(x,0)=\\sum_{n=1}^\\infty \\frac{-4}{n^{7} } \\sin(n \\pi x).$\nThen the solution is\n$ u(x,t)=\\sum_{n=1}^\\infty$ [ANS] $e^{-(n\\pi)^2t} \\sin(n\\pi x)$", "answer_v3": [ "-[4/(n^7)]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0298", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Heat equation", "level": "3", "keywords": [ "PDE", "partial", "differential", "equation" ], "problem_v1": "Suppose we have a laterally insulated metal rod of length 1, parametrized by length, starting with 0 at the left hand endpoint. Suppose the insulation is not perfect and there is lateral heat loss. Suppose left hand endpoint is kept at temperature-2 and right hand endpoint is perfectly insulated.\nWhich equation most likely governs this setup: [ANS] u_t=\\alpha^2 u_{xx}-8 t [ANS] u_t=\\alpha^2 u_{xx}-8 u_x [ANS] u_{tt}=\\alpha^2 u_{xx} [ANS] u_{tt}+u_{xx}=0 [ANS] u_t=\\alpha^2 u_{xx}-8 u [ANS] u_t=\\alpha^2 u_{xx} [ANS] u_t=\\alpha^2 u_{xx}-8x\nWhich boundary conditions are the correct ones: [ANS] u_x(0,t)=-2, \\quad u(1,t)=0 [ANS] u(0,t)=-2, \\quad u_x(1,t)=0 [ANS] u_x(0,t)=-2, \\quad u_x(1,t)=-2 [ANS] u(0,t)=-2, \\quad u(1,t)=0 [ANS] u_x(0,t)=-2, \\quad u_x(1,t)=0 [ANS] u(0,t)=0, \\quad u(1,t)=-2", "answer_v1": [ "Choice 5", "Choice 2" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Suppose we have a laterally insulated metal rod of length 1, parametrized by length, starting with 0 at the left hand endpoint. Suppose the insulation is not perfect and there is lateral heat loss. Suppose left hand endpoint is kept at temperature 1 and right hand endpoint is perfectly insulated.\nWhich equation most likely governs this setup: [ANS] u_t=\\alpha^2 u_{xx} [ANS] u_{tt}=\\alpha^2 u_{xx} [ANS] u_t=\\alpha^2 u_{xx}-2 u_x [ANS] u_{tt}+u_{xx}=0 [ANS] u_t=\\alpha^2 u_{xx}-2x [ANS] u_t=\\alpha^2 u_{xx}-2 u [ANS] u_t=\\alpha^2 u_{xx}-2 t\nWhich boundary conditions are the correct ones: [ANS] u(0,t)=1, \\quad u(1,t)=0 [ANS] u(0,t)=1, \\quad u_x(1,t)=0 [ANS] u_x(0,t)=1, \\quad u_x(1,t)=1 [ANS] u_x(0,t)=1, \\quad u(1,t)=0 [ANS] u(0,t)=0, \\quad u(1,t)=1 [ANS] u_x(0,t)=1, \\quad u_x(1,t)=0", "answer_v2": [ "Choice 6", "Choice 2" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Suppose we have a laterally insulated metal rod of length 1, parametrized by length, starting with 0 at the left hand endpoint. Suppose the insulation is not perfect and there is lateral heat loss. Suppose left hand endpoint is kept at temperature 7 and right hand endpoint is perfectly insulated.\nWhich equation most likely governs this setup: [ANS] u_t=\\alpha^2 u_{xx}-4 t [ANS] u_t=\\alpha^2 u_{xx} [ANS] u_t=\\alpha^2 u_{xx}-4 u_x [ANS] u_t=\\alpha^2 u_{xx}-4x [ANS] u_t=\\alpha^2 u_{xx}-4 u [ANS] u_{tt}=\\alpha^2 u_{xx} [ANS] u_{tt}+u_{xx}=0\nWhich boundary conditions are the correct ones: [ANS] u_x(0,t)=7, \\quad u_x(1,t)=0 [ANS] u(0,t)=7, \\quad u(1,t)=0 [ANS] u_x(0,t)=7, \\quad u(1,t)=0 [ANS] u(0,t)=0, \\quad u(1,t)=7 [ANS] u_x(0,t)=7, \\quad u_x(1,t)=7 [ANS] u(0,t)=7, \\quad u_x(1,t)=0", "answer_v3": [ "Choice 5", "Choice 6" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0299", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Heat equation", "level": "4", "keywords": [ "PDE", "partial", "differential", "equation", "heat equation", "convection" ], "problem_v1": "Solve the problem PDE: $\\quad u_t=u_{xx}-6 u_x, \\qquad-\\infty < x < \\infty, \\quad t > 0$ IC: $\\quad u(x,0)=8 e^{-(x^2)/4}$\nFirst, get rid of the convection term by changing variables, then use the Fourier transform. $u={}$ [ANS]", "answer_v1": [ "8/[sqrt(1+t)]*e^(-[(x-6*t)^2/[4*(1+t)]])" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the problem PDE: $\\quad u_t=u_{xx}-9 u_x, \\qquad-\\infty < x < \\infty, \\quad t > 0$ IC: $\\quad u(x,0)=2 e^{-(x^2)/4}$\nFirst, get rid of the convection term by changing variables, then use the Fourier transform. $u={}$ [ANS]", "answer_v2": [ "2/[sqrt(1+t)]*e^(-[(x-9*t)^2/[4*(1+t)]])" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the problem PDE: $\\quad u_t=u_{xx}-6 u_x, \\qquad-\\infty < x < \\infty, \\quad t > 0$ IC: $\\quad u(x,0)=4 e^{-(x^2)/4}$\nFirst, get rid of the convection term by changing variables, then use the Fourier transform. $u={}$ [ANS]", "answer_v3": [ "4/[sqrt(1+t)]*e^(-[(x-6*t)^2/[4*(1+t)]])" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0301", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Heat equation", "level": "4", "keywords": [ "PDE", "partial", "differential", "equation", "heat equation" ], "problem_v1": "The following scenario describes the temperature $u$ of a rod at position $x$ and time $t$. Consider the equation $u_t=u_{xx}$, $0 < x < 1$, $t > 0$, with boundary conditions $u(0,t)=0, u(1,t)=0$. Suppose $u(x,0)=\\sin(n \\pi x)$\nThat is, an input data of frequency $n$. Suppose $V(t)$ denotes the maximum variation of temperature on the rod at time $t$, that is, the maximum temperature minus the lowest temperature. We are interested as to what happens as $n$ grows. What is your intuition as to the behavior of $V(t)$: [ANS] A. $V(t)$ is independent of $n$. B. $V(t)$ is strictly decreasing. C. $V(t)$ is strictly increasing. D. $V(t)$ is constant. E. $V(t)$ goes to zero as $t \\to \\infty$. F. As $t \\to \\infty$, $V(t)$ goes to zero slower for larger $n$. G. As $t \\to \\infty$, $V(t)$ goes to zero faster for larger $n$. H. $V(t)$ goes to 1 as $t \\to \\infty$. I. None of the above", "answer_v1": [ "BEG" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ] ], "problem_v2": "The following scenario describes the temperature $u$ of a rod at position $x$ and time $t$. Consider the equation $u_t=u_{xx}$, $0 < x < 1$, $t > 0$, with boundary conditions $u(0,t)=0, u(1,t)=0$. Suppose $u(x,0)=\\sin(n \\pi x)$\nThat is, an input data of frequency $n$. Suppose $V(t)$ denotes the maximum variation of temperature on the rod at time $t$, that is, the maximum temperature minus the lowest temperature. We are interested as to what happens as $n$ grows. What is your intuition as to the behavior of $V(t)$: [ANS] A. $V(t)$ is strictly increasing. B. $V(t)$ goes to 1 as $t \\to \\infty$. C. $V(t)$ is strictly decreasing. D. $V(t)$ is constant. E. $V(t)$ goes to zero as $t \\to \\infty$. F. As $t \\to \\infty$, $V(t)$ goes to zero faster for larger $n$. G. $V(t)$ is independent of $n$. H. As $t \\to \\infty$, $V(t)$ goes to zero slower for larger $n$. I. None of the above", "answer_v2": [ "CEF" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ] ], "problem_v3": "The following scenario describes the temperature $u$ of a rod at position $x$ and time $t$. Consider the equation $u_t=u_{xx}$, $0 < x < 1$, $t > 0$, with boundary conditions $u(0,t)=0, u(1,t)=0$. Suppose $u(x,0)=\\sin(n \\pi x)$\nThat is, an input data of frequency $n$. Suppose $V(t)$ denotes the maximum variation of temperature on the rod at time $t$, that is, the maximum temperature minus the lowest temperature. We are interested as to what happens as $n$ grows. What is your intuition as to the behavior of $V(t)$: [ANS] A. $V(t)$ is strictly decreasing. B. $V(t)$ is constant. C. $V(t)$ goes to 1 as $t \\to \\infty$. D. $V(t)$ is independent of $n$. E. $V(t)$ goes to zero as $t \\to \\infty$. F. As $t \\to \\infty$, $V(t)$ goes to zero faster for larger $n$. G. $V(t)$ is strictly increasing. H. As $t \\to \\infty$, $V(t)$ goes to zero slower for larger $n$. I. None of the above", "answer_v3": [ "AEF" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ] ] }, { "id": "Differential_equations_0302", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Heat equation", "level": "3", "keywords": [ "PDE", "partial", "differential", "equation", "heat equation", "Laplace" ], "problem_v1": "We wish to solve the problem PDE: $\\quad w_t=8^2 w_{xx}, \\qquad-\\infty < x < \\infty, \\quad t > 0$ IC: $\\quad w(x,0)=7 \\sin(6x)$ using the Laplace transform.\nFirst if we let $W(x,s)=W(x)=\\mathcal{L}[w]$ with Laplace transforming the time variable, then the equation becomes (use only $W$ in your answer, not $W(x)$) [ANS] ${}={}$ [ANS] $ \\frac{d^2W}{dx^2} $\nThe solution then is $w(x,t)={}$ [ANS]", "answer_v1": [ "s*W-7*sin(6*x)", "8^2", "7*exp(-2304*t)*sin(6*x)" ], "answer_type_v1": [ "EX", "NV", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "We wish to solve the problem PDE: $\\quad w_t=2^2 w_{xx}, \\qquad-\\infty < x < \\infty, \\quad t > 0$ IC: $\\quad w(x,0)=3 \\sin(9x)$ using the Laplace transform.\nFirst if we let $W(x,s)=W(x)=\\mathcal{L}[w]$ with Laplace transforming the time variable, then the equation becomes (use only $W$ in your answer, not $W(x)$) [ANS] ${}={}$ [ANS] $ \\frac{d^2W}{dx^2} $\nThe solution then is $w(x,t)={}$ [ANS]", "answer_v2": [ "s*W-3*sin(9*x)", "2^2", "3*exp(-324*t)*sin(9*x)" ], "answer_type_v2": [ "EX", "NV", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "We wish to solve the problem PDE: $\\quad w_t=4^2 w_{xx}, \\qquad-\\infty < x < \\infty, \\quad t > 0$ IC: $\\quad w(x,0)=3 \\sin(6x)$ using the Laplace transform.\nFirst if we let $W(x,s)=W(x)=\\mathcal{L}[w]$ with Laplace transforming the time variable, then the equation becomes (use only $W$ in your answer, not $W(x)$) [ANS] ${}={}$ [ANS] $ \\frac{d^2W}{dx^2} $\nThe solution then is $w(x,t)={}$ [ANS]", "answer_v3": [ "s*W-3*sin(6*x)", "4^2", "3*exp(-576*t)*sin(6*x)" ], "answer_type_v3": [ "EX", "NV", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0304", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Heat equation", "level": "5", "keywords": [ "PDE", "partial", "differential", "equation", "heat equation" ], "problem_v1": "Suppose we have $u_t=a^2 u_{xx}$, $0 < x < 1, t > 0$, boundary conditions are $u(0,t)=u(1,t)=0$, and the initial condition is $u(x,0)=\\sin(\\pi x)$. What will be the behavior of $u(x,t)$ as time increases. There may be more than one correct answer. You do not need to solve the equation to answer this question. [ANS] A. The solution decreases slowly towards $u=0$. B. The solution decreases to minus infinity. C. The solution does not change as time goes on. D. The solution behaves unpredictably. E. The solution increases to infinity. F. For a fixed t, the solution will look kind of like an inverted parabola. G. For a fixed t, the solution will be linear H. The solution increases slowly towards $u=0$. I. For a fixed t, the second x derivative, $u_{xx}$, will take on both signs. J. For a fixed t, the second x derivative, $u_{xx}$, will always be negative. K. None of the above", "answer_v1": [ "AFJ" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ] ], "problem_v2": "Suppose we have $u_t=a^2 u_{xx}$, $0 < x < 1, t > 0$, boundary conditions are $u(0,t)=u(1,t)=0$, and the initial condition is $u(x,0)=\\sin(\\pi x)$. What will be the behavior of $u(x,t)$ as time increases. There may be more than one correct answer. You do not need to solve the equation to answer this question. [ANS] A. For a fixed t, the second x derivative, $u_{xx}$, will take on both signs. B. The solution does not change as time goes on. C. The solution decreases slowly towards $u=0$. D. For a fixed t, the solution will look kind of like an inverted parabola. E. The solution decreases to minus infinity. F. The solution increases to infinity. G. For a fixed t, the second x derivative, $u_{xx}$, will always be negative. H. The solution increases slowly towards $u=0$. I. For a fixed t, the solution will be linear J. The solution behaves unpredictably. K. None of the above", "answer_v2": [ "CDG" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ] ], "problem_v3": "Suppose we have $u_t=a^2 u_{xx}$, $0 < x < 1, t > 0$, boundary conditions are $u(0,t)=u(1,t)=0$, and the initial condition is $u(x,0)=\\sin(\\pi x)$. What will be the behavior of $u(x,t)$ as time increases. There may be more than one correct answer. You do not need to solve the equation to answer this question. [ANS] A. The solution increases slowly towards $u=0$. B. For a fixed t, the second x derivative, $u_{xx}$, will take on both signs. C. For a fixed t, the solution will be linear D. The solution decreases slowly towards $u=0$. E. The solution behaves unpredictably. F. The solution increases to infinity. G. For a fixed t, the solution will look kind of like an inverted parabola. H. The solution does not change as time goes on. I. For a fixed t, the second x derivative, $u_{xx}$, will always be negative. J. The solution decreases to minus infinity. K. None of the above", "answer_v3": [ "DGI" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ] ] }, { "id": "Differential_equations_0305", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Heat equation", "level": "5", "keywords": [ "PDE", "partial", "differential", "equation", "heat equation", "inhomogeneous" ], "problem_v1": "Consider the problem $u_t=u_{xx}$, $0 < x < 1$, $t > 0$, with boundary condition $u(0,t)=0, u(1,t)=\\cos(8 t)$, and initial condition $u(x,0)=x$.\nSolve in two steps. First transform the problem into a problem with homogeneous boundary conditions, but inhomogeneous PDE, and then solve. Write $u=S+U$, where $S(x,t)=A(t) (1-x)+B(t) x$ is (affine) linear in $x$, that is $S(x,t)={}$ [ANS]\nThe initial boundary value problem for $U$ becomes PDE: $U_t=U_{xx}+f(x,t)$ where $f(x,t)={}$ [ANS]\nBC: $U(0,t)=0, U(1,t)=0$ IC: $U(x,0)={}$ [ANS]\nNow solve for $U$.\nThe eigenfunctions to use are $X_n(x)={}$ [ANS]\nFirst decompose the inhomogeneity in the PDE as $ f(x,t)=\\sum_{j=1}^n f_n(t) X_n(x)$, where $f_n(t)={}$ [ANS]\nYou find that $ U(x,t)=\\sum_{j=1}^\\infty T_n(t) X_n(x)$, where $T_n={}$ $ \\int_0^t$ [ANS] $ds$\nThen $u(x,t)=S(x,t)+U(x,t)$", "answer_v1": [ "x*cos(8*t)", "8*x*sin(8*t)", "0", "sin(n*pi*x)", "16*sin(8*t)*(-1)^{n+1}/(pi*n)", "8*2*e^[-(t-s)*(n*pi)^2]*sin(8*s)*(-1)^{n+1}/(pi*n)" ], "answer_type_v1": [ "EX", "EX", "NV", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Consider the problem $u_t=u_{xx}$, $0 < x < 1$, $t > 0$, with boundary condition $u(0,t)=0, u(1,t)=\\cos(2 t)$, and initial condition $u(x,0)=x$.\nSolve in two steps. First transform the problem into a problem with homogeneous boundary conditions, but inhomogeneous PDE, and then solve. Write $u=S+U$, where $S(x,t)=A(t) (1-x)+B(t) x$ is (affine) linear in $x$, that is $S(x,t)={}$ [ANS]\nThe initial boundary value problem for $U$ becomes PDE: $U_t=U_{xx}+f(x,t)$ where $f(x,t)={}$ [ANS]\nBC: $U(0,t)=0, U(1,t)=0$ IC: $U(x,0)={}$ [ANS]\nNow solve for $U$.\nThe eigenfunctions to use are $X_n(x)={}$ [ANS]\nFirst decompose the inhomogeneity in the PDE as $ f(x,t)=\\sum_{j=1}^n f_n(t) X_n(x)$, where $f_n(t)={}$ [ANS]\nYou find that $ U(x,t)=\\sum_{j=1}^\\infty T_n(t) X_n(x)$, where $T_n={}$ $ \\int_0^t$ [ANS] $ds$\nThen $u(x,t)=S(x,t)+U(x,t)$", "answer_v2": [ "x*cos(2*t)", "2*x*sin(2*t)", "0", "sin(n*pi*x)", "4*sin(2*t)*(-1)^{n+1}/(pi*n)", "2*2*e^[-(t-s)*(n*pi)^2]*sin(2*s)*(-1)^{n+1}/(pi*n)" ], "answer_type_v2": [ "EX", "EX", "NV", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Consider the problem $u_t=u_{xx}$, $0 < x < 1$, $t > 0$, with boundary condition $u(0,t)=0, u(1,t)=\\cos(4 t)$, and initial condition $u(x,0)=x$.\nSolve in two steps. First transform the problem into a problem with homogeneous boundary conditions, but inhomogeneous PDE, and then solve. Write $u=S+U$, where $S(x,t)=A(t) (1-x)+B(t) x$ is (affine) linear in $x$, that is $S(x,t)={}$ [ANS]\nThe initial boundary value problem for $U$ becomes PDE: $U_t=U_{xx}+f(x,t)$ where $f(x,t)={}$ [ANS]\nBC: $U(0,t)=0, U(1,t)=0$ IC: $U(x,0)={}$ [ANS]\nNow solve for $U$.\nThe eigenfunctions to use are $X_n(x)={}$ [ANS]\nFirst decompose the inhomogeneity in the PDE as $ f(x,t)=\\sum_{j=1}^n f_n(t) X_n(x)$, where $f_n(t)={}$ [ANS]\nYou find that $ U(x,t)=\\sum_{j=1}^\\infty T_n(t) X_n(x)$, where $T_n={}$ $ \\int_0^t$ [ANS] $ds$\nThen $u(x,t)=S(x,t)+U(x,t)$", "answer_v3": [ "x*cos(4*t)", "4*x*sin(4*t)", "0", "sin(n*pi*x)", "8*sin(4*t)*(-1)^{n+1}/(pi*n)", "4*2*e^[-(t-s)*(n*pi)^2]*sin(4*s)*(-1)^{n+1}/(pi*n)" ], "answer_type_v3": [ "EX", "EX", "NV", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Differential_equations_0307", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Heat equation", "level": "5", "keywords": [ "PDE", "non-homogeneous", "boundary", "value" ], "problem_v1": "For partial derivatives of a function use the subscript notation; so for the second partial derivative of the function u(x,t) with respect to x use uxx. For ordinary differential equations use the prime notation, so the second derivative of the function f(x) is f''.\nSolve the heat equation\n$ k \\frac{\\partial^2 u}{\\partial x^2} +12= \\frac{\\partial u}{\\partial t} , 0< x < 5, t > 0$\n$ u(0,t)=103, u(5,t)=98, t > 0$\n$ u(x,0)=x\\!\\left(5-x\\right), 0 < x < 5$ using a steady-state and transient solution: ie write $u(x,t)=v(x,t)+S(x)$ with $v$ a solution of the heat equation with homogeneous boundary conditions\n$ k \\frac{\\partial^2 v}{\\partial x^2} = \\frac{\\partial v}{\\partial t} , 0< x < 5, t > 0$\n$ v(0,t)=0, v(5,t)=0, t > 0$\n$ v(x,0)=x\\!\\left(5-x\\right)-S(x), 0 < x < 5$\nUsing the substitution $u=v+S$\n$ k \\frac{\\partial^2 u}{\\partial x^2} $=[ANS]\n$ \\frac{\\partial u}{\\partial t} $=[ANS]\nSo the steady-state solution S must satisfy the IVP (DE first, then IC's) [ANS] $=0$\n$ S(0)=$ [ANS]\n$ S(5)=$ [ANS]\nTherefore $S(x)=$ [ANS]\nand $v(x,t)= \\frac{2}{5} \\sum\\limits_{n=1}^\\infty b_n e^{\\left(-k \\frac{n^2\\pi^2}{{5} ^2}t\\right)} \\sin\\left( \\frac{n\\pi}{5} x\\right)$ with $b_n=\\int_0^{5}$ [ANS] $ dx$\nTherefore the solution of nonhomogeneous heat equation is $u(x,t)=v+S=$ $ \\frac{2}{5} \\sum\\limits_{n=1}^\\infty\\Bigg($ [ANS] $e^{\\left(-k \\frac{n^2\\pi^2}{{5} ^2}t\\right)} \\sin\\left( \\frac{n\\pi}{5} x\\right)\\Bigg)+$ [ANS]", "answer_v1": [ "k*(vxx+S", "vt", "k*S", "103", "98", "-12/(2*k)*x^2+[(-5)+12/(2*k)*25]/5*x+103", "[x*(5-x)-[-12/(2*k)*x^2+[-5+12/(2*k)*25]/5*x+103]]*sin(n*pi*x/5)", "5/(2*k*n^3*pi^3)*[4*k*25-600-2*k*n^2*pi^2*103+(-4*k*25+600+2*k*n^2*pi^2*98)*(-1)^n]", "-12/(2*k)*x^2+[(-5)+12/(2*k)*25]/5*x+103" ], "answer_type_v1": [ "EX", "EX", "EX", "NV", "NV", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "For partial derivatives of a function use the subscript notation; so for the second partial derivative of the function u(x,t) with respect to x use uxx. For ordinary differential equations use the prime notation, so the second derivative of the function f(x) is f''.\nSolve the heat equation\n$ k \\frac{\\partial^2 u}{\\partial x^2} +8= \\frac{\\partial u}{\\partial t} , 0< x < 2, t > 0$\n$ u(0,t)=82, u(2,t)=108, t > 0$\n$ u(x,0)=x\\!\\left(2-x\\right), 0 < x < 2$ using a steady-state and transient solution: ie write $u(x,t)=v(x,t)+S(x)$ with $v$ a solution of the heat equation with homogeneous boundary conditions\n$ k \\frac{\\partial^2 v}{\\partial x^2} = \\frac{\\partial v}{\\partial t} , 0< x < 2, t > 0$\n$ v(0,t)=0, v(2,t)=0, t > 0$\n$ v(x,0)=x\\!\\left(2-x\\right)-S(x), 0 < x < 2$\nUsing the substitution $u=v+S$\n$ k \\frac{\\partial^2 u}{\\partial x^2} $=[ANS]\n$ \\frac{\\partial u}{\\partial t} $=[ANS]\nSo the steady-state solution S must satisfy the IVP (DE first, then IC's) [ANS] $=0$\n$ S(0)=$ [ANS]\n$ S(2)=$ [ANS]\nTherefore $S(x)=$ [ANS]\nand $v(x,t)= \\frac{2}{2} \\sum\\limits_{n=1}^\\infty b_n e^{\\left(-k \\frac{n^2\\pi^2}{{2} ^2}t\\right)} \\sin\\left( \\frac{n\\pi}{2} x\\right)$ with $b_n=\\int_0^{2}$ [ANS] $ dx$\nTherefore the solution of nonhomogeneous heat equation is $u(x,t)=v+S=$ $ \\frac{2}{2} \\sum\\limits_{n=1}^\\infty\\Bigg($ [ANS] $e^{\\left(-k \\frac{n^2\\pi^2}{{2} ^2}t\\right)} \\sin\\left( \\frac{n\\pi}{2} x\\right)\\Bigg)+$ [ANS]", "answer_v2": [ "k*(vxx+S", "vt", "k*S", "82", "108", "-8/(2*k)*x^2+[26+8/(2*k)*4]/2*x+82", "[x*(2-x)-[-8/(2*k)*x^2+[26+8/(2*k)*4]/2*x+82]]*sin(n*pi*x/2)", "2/(2*k*n^3*pi^3)*[4*k*4-64-2*k*n^2*pi^2*82+(-4*k*4+64+2*k*n^2*pi^2*108)*(-1)^n]", "-8/(2*k)*x^2+[26+8/(2*k)*4]/2*x+82" ], "answer_type_v2": [ "EX", "EX", "EX", "NV", "NV", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "For partial derivatives of a function use the subscript notation; so for the second partial derivative of the function u(x,t) with respect to x use uxx. For ordinary differential equations use the prime notation, so the second derivative of the function f(x) is f''.\nSolve the heat equation\n$ k \\frac{\\partial^2 u}{\\partial x^2} +11= \\frac{\\partial u}{\\partial t} , 0< x < 3, t > 0$\n$ u(0,t)=89, u(3,t)=98, t > 0$\n$ u(x,0)=x\\!\\left(3-x\\right), 0 < x < 3$ using a steady-state and transient solution: ie write $u(x,t)=v(x,t)+S(x)$ with $v$ a solution of the heat equation with homogeneous boundary conditions\n$ k \\frac{\\partial^2 v}{\\partial x^2} = \\frac{\\partial v}{\\partial t} , 0< x < 3, t > 0$\n$ v(0,t)=0, v(3,t)=0, t > 0$\n$ v(x,0)=x\\!\\left(3-x\\right)-S(x), 0 < x < 3$\nUsing the substitution $u=v+S$\n$ k \\frac{\\partial^2 u}{\\partial x^2} $=[ANS]\n$ \\frac{\\partial u}{\\partial t} $=[ANS]\nSo the steady-state solution S must satisfy the IVP (DE first, then IC's) [ANS] $=0$\n$ S(0)=$ [ANS]\n$ S(3)=$ [ANS]\nTherefore $S(x)=$ [ANS]\nand $v(x,t)= \\frac{2}{3} \\sum\\limits_{n=1}^\\infty b_n e^{\\left(-k \\frac{n^2\\pi^2}{{3} ^2}t\\right)} \\sin\\left( \\frac{n\\pi}{3} x\\right)$ with $b_n=\\int_0^{3}$ [ANS] $ dx$\nTherefore the solution of nonhomogeneous heat equation is $u(x,t)=v+S=$ $ \\frac{2}{3} \\sum\\limits_{n=1}^\\infty\\Bigg($ [ANS] $e^{\\left(-k \\frac{n^2\\pi^2}{{3} ^2}t\\right)} \\sin\\left( \\frac{n\\pi}{3} x\\right)\\Bigg)+$ [ANS]", "answer_v3": [ "k*(vxx+S", "vt", "k*S", "89", "98", "-11/(2*k)*x^2+[9+11/(2*k)*9]/3*x+89", "[x*(3-x)-[-11/(2*k)*x^2+[9+11/(2*k)*9]/3*x+89]]*sin(n*pi*x/3)", "3/(2*k*n^3*pi^3)*[4*k*9-198-2*k*n^2*pi^2*89+(-4*k*9+198+2*k*n^2*pi^2*98)*(-1)^n]", "-11/(2*k)*x^2+[9+11/(2*k)*9]/3*x+89" ], "answer_type_v3": [ "EX", "EX", "EX", "NV", "NV", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Differential_equations_0308", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Heat equation", "level": "4", "keywords": [ "heat" ], "problem_v1": "The temperature in a rod of unit length in which there is heat transfer from its right boundary into a surrounding medium kept at a constant temperature zero is determined from\n$ {\\rm k} \\frac{\\partial^2u}{\\partial x^2} = \\frac{\\partial u}{\\partial t} $ $ u\\left(0,t\\right)=0, \\frac{\\partial u}{\\partial x} \\Bigg\\vert_{x=1}=-0.7 u\\left(1,t\\right), t > 0$ $ u\\left(x,0\\right)=0, 0 < x < 1$\nTherefore $u(x,t)=1.4\\sum\\limits_{n=1}^\\infty \\frac{1-\\cos(\\alpha_n)}{\\alpha_n\\left(0.7+\\cos^2(\\alpha_n)\\right)} e^{-k\\alpha_n^2t}\\sin(\\alpha_n x)$ where $\\tan(\\alpha_n)=- \\frac{\\alpha_n}{0.7} $. Fill in the table below with the first four values of $\\alpha_n$:\n$\\begin{array}{cc}\\hline n & \\alpha_n \\\\ \\hline 1 & [ANS] \\\\ \\hline 2 & [ANS] \\\\ \\hline 3 & [ANS] \\\\ \\hline 4 & [ANS] \\\\ \\hline \\end{array}$\nHint: solve tan(x)=-x/h from 0 to a in wolframalpha works, you need to supply h and a.", "answer_v1": [ "0", "1.92035", "4.85557", "7.94189" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The temperature in a rod of unit length in which there is heat transfer from its right boundary into a surrounding medium kept at a constant temperature zero is determined from\n$ {\\rm k} \\frac{\\partial^2u}{\\partial x^2} = \\frac{\\partial u}{\\partial t} $ $ u\\left(0,t\\right)=0, \\frac{\\partial u}{\\partial x} \\Bigg\\vert_{x=1}=-0.1 u\\left(1,t\\right), t > 0$ $ u\\left(x,0\\right)=0, 0 < x < 1$\nTherefore $u(x,t)=0.2\\sum\\limits_{n=1}^\\infty \\frac{1-\\cos(\\alpha_n)}{\\alpha_n\\left(0.1+\\cos^2(\\alpha_n)\\right)} e^{-k\\alpha_n^2t}\\sin(\\alpha_n x)$ where $\\tan(\\alpha_n)=- \\frac{\\alpha_n}{0.1} $. Fill in the table below with the first four values of $\\alpha_n$:\n$\\begin{array}{cc}\\hline n & \\alpha_n \\\\ \\hline 1 & [ANS] \\\\ \\hline 2 & [ANS] \\\\ \\hline 3 & [ANS] \\\\ \\hline 4 & [ANS] \\\\ \\hline \\end{array}$\nHint: solve tan(x)=-x/h from 0 to a in wolframalpha works, you need to supply h and a.", "answer_v2": [ "0", "1.63199", "4.73351", "7.86669" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The temperature in a rod of unit length in which there is heat transfer from its right boundary into a surrounding medium kept at a constant temperature zero is determined from\n$ {\\rm k} \\frac{\\partial^2u}{\\partial x^2} = \\frac{\\partial u}{\\partial t} $ $ u\\left(0,t\\right)=0, \\frac{\\partial u}{\\partial x} \\Bigg\\vert_{x=1}=-0.3 u\\left(1,t\\right), t > 0$ $ u\\left(x,0\\right)=0, 0 < x < 1$\nTherefore $u(x,t)=0.6\\sum\\limits_{n=1}^\\infty \\frac{1-\\cos(\\alpha_n)}{\\alpha_n\\left(0.3+\\cos^2(\\alpha_n)\\right)} e^{-k\\alpha_n^2t}\\sin(\\alpha_n x)$ where $\\tan(\\alpha_n)=- \\frac{\\alpha_n}{0.3} $. Fill in the table below with the first four values of $\\alpha_n$:\n$\\begin{array}{cc}\\hline n & \\alpha_n \\\\ \\hline 1 & [ANS] \\\\ \\hline 2 & [ANS] \\\\ \\hline 3 & [ANS] \\\\ \\hline 4 & [ANS] \\\\ \\hline \\end{array}$\nHint: solve tan(x)=-x/h from 0 to a in wolframalpha works, you need to supply h and a.", "answer_v3": [ "0", "1.7414", "4.77513", "7.89198" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0309", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Heat equation", "level": "5", "keywords": [ "PDE", "heat", "equation", "dimensional" ], "problem_v1": "Suppose the rectangular region $0 < x < 6, 0 < y < 5$ is a thin plate in which the temperature u is a function of time t and position (x,y). If the initial temperature is f(x,y) and the boundaries are held at temperature for $t > 0$ then u satisfies the two-dimensional heat equation k\\left( \\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} \\right)= \\frac{\\partial u}{\\partial t} , 0< x < 6, 0 0 u\\left(0,y,t\\right)=0, u\\left(6,y,t\\right)=0, 00 u\\left(x,0,t\\right)=0, u\\left(x,5,t\\right)=0, 00 u\\left(x,y,0\\right)=\\left(x-6\\right)\\!\\left(y-5\\right), 0 0$ then u satisfies the two-dimensional heat equation k\\left( \\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} \\right)= \\frac{\\partial u}{\\partial t} , 0< x < 1, 0 0 u\\left(0,y,t\\right)=0, u\\left(1,y,t\\right)=0, 00 u\\left(x,0,t\\right)=0, u\\left(x,7,t\\right)=0, 00 u\\left(x,y,0\\right)=\\left(x-1\\right)\\!\\left(y-7\\right), 0 0$ then u satisfies the two-dimensional heat equation k\\left( \\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} \\right)= \\frac{\\partial u}{\\partial t} , 0< x < 3, 0 0 u\\left(0,y,t\\right)=0, u\\left(3,y,t\\right)=0, 00 u\\left(x,0,t\\right)=0, u\\left(x,5,t\\right)=0, 00 u\\left(x,y,0\\right)=\\left(x-3\\right)\\!\\left(y-5\\right), 00$ C. $ k \\frac{\\partial^2 u}{\\partial x^2} = \\frac{\\partial u}{\\partial t} , \\quad 00$\nBoundary/Initial Conditions [ANS] A. $\\begin{aligned} u(0,t) &=0 & \\frac{\\partial u}{\\partial x} \\Bigg\\vert_{x=L} &=0, &t > 0\\\\ u(x,0) &=f(x),&& &0 < x < L\\\\ &&&&\\\\ &&&&\\\\ \\end{aligned}$ B. $\\begin{aligned} \\frac{\\partial u}{\\partial x} \\Bigg\\vert_{x=0} &=0, & \\frac{\\partial u}{\\partial x} \\Bigg\\vert_{x=L} &=0, &0 < y < b\\\\ u(x,0) &=0, &u(x,b) &=f(x), &0 < x < L\\\\ &&&&\\\\ &&&&\\\\ \\end{aligned}$ C. $\\begin{aligned} u(0,t) &=0 &u(L,t) &=0, &t > 0\\\\ u(x,0) &=f(x),& \\frac{\\partial u}{\\partial t} \\Bigg\\vert_{t=0} &=0, &0 < x < L\\\\ &&&&\\\\ &&&&\\\\ \\end{aligned}$", "answer_v1": [ "C", "A" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v2": "The left end of a rod of length L is held at temperature 0, and the right end is insulated. The initial temperature is f(x) throughout. Choose the PDE and boundary/initial conditions that model this scenario.\nPartial Differential Equation [ANS] A. $ k \\frac{\\partial^2 u}{\\partial x^2} = \\frac{\\partial u}{\\partial t} , \\quad 00$ B. $\\alpha^2 \\frac{\\partial^2 u}{\\partial x^2} = \\frac{\\partial^2 u}{\\partial t^2} , \\quad 00$ C. $ \\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} =0, \\quad 0 0\\\\ u(x,0) &=f(x),&& &0 < x < L\\\\ &&&&\\\\ &&&&\\\\ \\end{aligned}$ B. $\\begin{aligned} u(0,t) &=0 &u(L,t) &=0, &t > 0\\\\ u(x,0) &=f(x),& \\frac{\\partial u}{\\partial t} \\Bigg\\vert_{t=0} &=0, &0 < x < L\\\\ &&&&\\\\ &&&&\\\\ \\end{aligned}$ C. $\\begin{aligned} \\frac{\\partial u}{\\partial x} \\Bigg\\vert_{x=0} &=0, & \\frac{\\partial u}{\\partial x} \\Bigg\\vert_{x=L} &=0, &0 < y < b\\\\ u(x,0) &=0, &u(x,b) &=f(x), &0 < x < L\\\\ &&&&\\\\ &&&&\\\\ \\end{aligned}$", "answer_v2": [ "A", "A" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "The left end of a rod of length L is held at temperature 0, and the right end is insulated. The initial temperature is f(x) throughout. Choose the PDE and boundary/initial conditions that model this scenario.\nPartial Differential Equation [ANS] A. $ \\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} =0, \\quad 00$ C. $\\alpha^2 \\frac{\\partial^2 u}{\\partial x^2} = \\frac{\\partial^2 u}{\\partial t^2} , \\quad 00$\nBoundary/Initial Conditions [ANS] A. $\\begin{aligned} \\frac{\\partial u}{\\partial x} \\Bigg\\vert_{x=0} &=0, & \\frac{\\partial u}{\\partial x} \\Bigg\\vert_{x=L} &=0, &0 < y < b\\\\ u(x,0) &=0, &u(x,b) &=f(x), &0 < x < L\\\\ &&&&\\\\ &&&&\\\\ \\end{aligned}$ B. $\\begin{aligned} u(0,t) &=0 & \\frac{\\partial u}{\\partial x} \\Bigg\\vert_{x=L} &=0, &t > 0\\\\ u(x,0) &=f(x),&& &0 < x < L\\\\ &&&&\\\\ &&&&\\\\ \\end{aligned}$ C. $\\begin{aligned} u(0,t) &=0 &u(L,t) &=0, &t > 0\\\\ u(x,0) &=f(x),& \\frac{\\partial u}{\\partial t} \\Bigg\\vert_{t=0} &=0, &0 < x < L\\\\ &&&&\\\\ &&&&\\\\ \\end{aligned}$", "answer_v3": [ "B", "B" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C" ], [ "A", "B", "C" ] ] }, { "id": "Differential_equations_0313", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Wave equation", "level": "4", "keywords": [ "PDE", "partial", "differential", "equation", "wave equation", "planar wave" ], "problem_v1": "Suppose we have the 3D wave equation $u_{tt}=64 (u_{xx}+u_{yy}+u_{zz})$ $-\\infty < x,y,z < \\infty$ and $t > 0$. Also suppose for simplicity that $u_t(x,y,z,0)=0$ is one of the initial conditions.\na) Find the solution of the equation if the other initial condition is $u(x,y,z,0)=\\sin(x)$ $u(x,y,z,t)={}$ [ANS]\nHint: Planar waves and 1D wave equation.\nb) Find the solution of the equation if the other initial condition is $u(x,y,z,0)=\\cos(y)$ $u(x,y,z,t)={}$ [ANS]\nc) Find the solution of the equation if the other initial condition is $u(x,y,z,0)=\\cos(6 z)$ $u(x,y,z,t)={}$ [ANS]\nd) Find the solution of the equation if the other initial condition is $u(x,y,z,0)=\\sin(x)+7 \\cos(y)+4 \\cos(6 z)$ $u(x,y,z,t)={}$ [ANS]", "answer_v1": [ "[sin(x-8*t)+sin(x+8*t)]/2", "[cos(y-8*t)+cos(y+8*t)]/2", "[cos(6*(z-8*t))+cos(6*(z+8*t))]/2", "[sin(x-8*t)+sin(x+8*t)]/2+7*[cos(y-8*t)+cos(y+8*t)]/2+4*[cos(6*(z-8*t))+cos(6*(z+8*t))]/2" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose we have the 3D wave equation $u_{tt}=4 (u_{xx}+u_{yy}+u_{zz})$ $-\\infty < x,y,z < \\infty$ and $t > 0$. Also suppose for simplicity that $u_t(x,y,z,0)=0$ is one of the initial conditions.\na) Find the solution of the equation if the other initial condition is $u(x,y,z,0)=\\sin(x)$ $u(x,y,z,t)={}$ [ANS]\nHint: Planar waves and 1D wave equation.\nb) Find the solution of the equation if the other initial condition is $u(x,y,z,0)=\\cos(y)$ $u(x,y,z,t)={}$ [ANS]\nc) Find the solution of the equation if the other initial condition is $u(x,y,z,0)=\\cos(9 z)$ $u(x,y,z,t)={}$ [ANS]\nd) Find the solution of the equation if the other initial condition is $u(x,y,z,0)=\\sin(x)+3 \\cos(y)+4 \\cos(9 z)$ $u(x,y,z,t)={}$ [ANS]", "answer_v2": [ "[sin(x-2*t)+sin(x+2*t)]/2", "[cos(y-2*t)+cos(y+2*t)]/2", "[cos(9*(z-2*t))+cos(9*(z+2*t))]/2", "[sin(x-2*t)+sin(x+2*t)]/2+3*[cos(y-2*t)+cos(y+2*t)]/2+4*[cos(9*(z-2*t))+cos(9*(z+2*t))]/2" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose we have the 3D wave equation $u_{tt}=16 (u_{xx}+u_{yy}+u_{zz})$ $-\\infty < x,y,z < \\infty$ and $t > 0$. Also suppose for simplicity that $u_t(x,y,z,0)=0$ is one of the initial conditions.\na) Find the solution of the equation if the other initial condition is $u(x,y,z,0)=\\sin(x)$ $u(x,y,z,t)={}$ [ANS]\nHint: Planar waves and 1D wave equation.\nb) Find the solution of the equation if the other initial condition is $u(x,y,z,0)=\\cos(y)$ $u(x,y,z,t)={}$ [ANS]\nc) Find the solution of the equation if the other initial condition is $u(x,y,z,0)=\\cos(6 z)$ $u(x,y,z,t)={}$ [ANS]\nd) Find the solution of the equation if the other initial condition is $u(x,y,z,0)=\\sin(x)+3 \\cos(y)+8 \\cos(6 z)$ $u(x,y,z,t)={}$ [ANS]", "answer_v3": [ "[sin(x-4*t)+sin(x+4*t)]/2", "[cos(y-4*t)+cos(y+4*t)]/2", "[cos(6*(z-4*t))+cos(6*(z+4*t))]/2", "[sin(x-4*t)+sin(x+4*t)]/2+3*[cos(y-4*t)+cos(y+4*t)]/2+8*[cos(6*(z-4*t))+cos(6*(z+4*t))]/2" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Differential_equations_0314", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Wave equation", "level": "4", "keywords": [ "PDE", "partial", "differential", "equation", "wave equation" ], "problem_v1": "Take the vibrating string satisfying PDE: $\\quad u_{tt}=\\alpha^2 u_{xx}, \\qquad 0 < x < L, \\quad t > 0$ BC: $\\quad u(0,t)=u(L,t)=0$ The constant $\\alpha^2= \\frac{T}{\\rho} $ where $T$ is tension and $\\rho$ is linear density of the string.\nSuppose the string vibrates at base frequency $\\omega$.\na) If we lengthen the string to $8 L$ the frequency $\\omega$ gets multiplied by [ANS].\nb) If we increase tension by a factor of $36$ the frequency $\\omega$ gets multiplied by [ANS].", "answer_v1": [ "1/8", "6" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Take the vibrating string satisfying PDE: $\\quad u_{tt}=\\alpha^2 u_{xx}, \\qquad 0 < x < L, \\quad t > 0$ BC: $\\quad u(0,t)=u(L,t)=0$ The constant $\\alpha^2= \\frac{T}{\\rho} $ where $T$ is tension and $\\rho$ is linear density of the string.\nSuppose the string vibrates at base frequency $\\omega$.\na) If we lengthen the string to $2 L$ the frequency $\\omega$ gets multiplied by [ANS].\nb) If we increase tension by a factor of $81$ the frequency $\\omega$ gets multiplied by [ANS].", "answer_v2": [ "1/2", "9" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Take the vibrating string satisfying PDE: $\\quad u_{tt}=\\alpha^2 u_{xx}, \\qquad 0 < x < L, \\quad t > 0$ BC: $\\quad u(0,t)=u(L,t)=0$ The constant $\\alpha^2= \\frac{T}{\\rho} $ where $T$ is tension and $\\rho$ is linear density of the string.\nSuppose the string vibrates at base frequency $\\omega$.\na) If we lengthen the string to $4 L$ the frequency $\\omega$ gets multiplied by [ANS].\nb) If we increase tension by a factor of $36$ the frequency $\\omega$ gets multiplied by [ANS].", "answer_v3": [ "1/4", "6" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0315", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Wave equation", "level": "4", "keywords": [ "PDE", "partial", "differential", "equation", "wave equation" ], "problem_v1": "Suppose we have the 3D wave equation $u_{tt}=64 (u_{xx}+u_{yy}+u_{zz})$ $-\\infty < x,y,z < \\infty$ and $t > 0$. Suppose that $u(x,y,z,0)=f(x,y,z)$ $u_t(x,y,z,0)=g(x,y,z)$\nWhere $f(x,y,z)$ and $g(x,y,z)$ are zero outside the sphere of radius 6, and nonzero inside the sphere. At what time will an observer at position $(42,0,0)$ notice the disturbance, that is, what is the first time that $u(x,y,z,t)$ will be nonzero. $t={}$ [ANS]", "answer_v1": [ "(42-6)/8" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose we have the 3D wave equation $u_{tt}=4 (u_{xx}+u_{yy}+u_{zz})$ $-\\infty < x,y,z < \\infty$ and $t > 0$. Suppose that $u(x,y,z,0)=f(x,y,z)$ $u_t(x,y,z,0)=g(x,y,z)$\nWhere $f(x,y,z)$ and $g(x,y,z)$ are zero outside the sphere of radius 9, and nonzero inside the sphere. At what time will an observer at position $(27,0,0)$ notice the disturbance, that is, what is the first time that $u(x,y,z,t)$ will be nonzero. $t={}$ [ANS]", "answer_v2": [ "(27-9)/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose we have the 3D wave equation $u_{tt}=16 (u_{xx}+u_{yy}+u_{zz})$ $-\\infty < x,y,z < \\infty$ and $t > 0$. Suppose that $u(x,y,z,0)=f(x,y,z)$ $u_t(x,y,z,0)=g(x,y,z)$\nWhere $f(x,y,z)$ and $g(x,y,z)$ are zero outside the sphere of radius 6, and nonzero inside the sphere. At what time will an observer at position $(18,0,0)$ notice the disturbance, that is, what is the first time that $u(x,y,z,t)$ will be nonzero. $t={}$ [ANS]", "answer_v3": [ "(18-6)/4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0316", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Wave equation", "level": "4", "keywords": [ "PDE", "partial", "differential", "equation", "wave equation" ], "problem_v1": "We wish to solve the PDE $u_{tt}=64 u_{xx}, \\qquad-\\infty < x < \\infty, \\quad t > 0$\nGuess solutions of the form $u=e^{\\alpha x+\\beta t}$ (find a relationship between $\\alpha$ and $\\beta$).\nSuppose you know $u(x,0)=e^{x}$. There are exactly two solutions of the above form, one with a positive $u_t(x,0)$ and one with negative $u_t(x,0)$. The first is: $u={}$ [ANS]\nAnd the second is $u={}$ [ANS]", "answer_v1": [ "e^(x+8*t)", "e^(x-8*t)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "We wish to solve the PDE $u_{tt}=4 u_{xx}, \\qquad-\\infty < x < \\infty, \\quad t > 0$\nGuess solutions of the form $u=e^{\\alpha x+\\beta t}$ (find a relationship between $\\alpha$ and $\\beta$).\nSuppose you know $u(x,0)=e^{x}$. There are exactly two solutions of the above form, one with a positive $u_t(x,0)$ and one with negative $u_t(x,0)$. The first is: $u={}$ [ANS]\nAnd the second is $u={}$ [ANS]", "answer_v2": [ "e^(x+2*t)", "e^(x-2*t)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "We wish to solve the PDE $u_{tt}=16 u_{xx}, \\qquad-\\infty < x < \\infty, \\quad t > 0$\nGuess solutions of the form $u=e^{\\alpha x+\\beta t}$ (find a relationship between $\\alpha$ and $\\beta$).\nSuppose you know $u(x,0)=e^{x}$. There are exactly two solutions of the above form, one with a positive $u_t(x,0)$ and one with negative $u_t(x,0)$. The first is: $u={}$ [ANS]\nAnd the second is $u={}$ [ANS]", "answer_v3": [ "e^(x+4*t)", "e^(x-4*t)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Differential_equations_0318", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Wave equation", "level": "5", "keywords": [ "PDE", "wave", "equation" ], "problem_v1": "For the general solution of the differential equation in X use A and B for your constants and list the functions in alphabetical order, for example $y=A\\cos (x)+B \\sin(x)$. For the differential equation in T use the C and D.For the variable $\\lambda$ type the word lambda and type alpha for $\\alpha$,otherwise treat them as you would any other variable.\nUse the prime notation for derivatives, so the derivative of $X$ is written as $X^\\prime$. Do NOT use $X^\\prime(x)$\nThe longitudinal displacement u(x,t) of a vibrating elastic bar can be modeled by a wave equation with free-end conditions\n a^2 \\frac{\\partial^2 u}{\\partial x^2} = \\frac{\\partial^2 u}{\\partial t^2} , 0< x < L, t > 0 \\frac{\\partial u}{\\partial x} \\Big\\vert_{x=0}=0, \\frac{\\partial u}{\\partial x} \\Big\\vert_{x=L}=0, t > 0 u(x,0)=x \\frac{\\partial u}{\\partial t} \\Big\\vert_{t=0}=2,0 0 \\frac{\\partial u}{\\partial x} \\Big\\vert_{x=0}=0, \\frac{\\partial u}{\\partial x} \\Big\\vert_{x=L}=0, t > 0 To add in the initial conditions we will have to use a series solution. For the series we relabel the coefficients so that $C=A_n$ and $D=B_n$ $u(x,t)=\\sum\\limits_{n=0}^\\infty \\Bigg(A_n$ [ANS] $+B_n$ [ANS] $\\Bigg)$ [ANS]\nUsing the initial conditions we can compute\n$\\begin{array}{ccccccccccccc}\\hline A_0 & &=& & \\frac{[ANS]}{[ANS]} & & \\int & & L 0 & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccc}\\hline & &=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccccc}\\hline A_n & &=& & \\frac{[ANS]}{[ANS]} & & \\int & & L 0 & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccc}\\hline & &=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccccc}\\hline B_n & &=& & \\frac{[ANS]}{[ANS]} & & \\int & & L 0 & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccc}\\hline & &=& & [ANS] \\\\ \\hline \\end{array}$\nPART 5: Finally!\n$u(x,t)=$ [ANS]+$\\sum\\limits_{n=1}^\\infty$ [ANS]", "answer_v1": [ "a^2*X", "X*T", "X", "T", "X", "0", "0", "L", "0", "T", "A+B*x", "B", "B", "A*cosh(alpha*x)+B*sinh(alpha*x)", "A*alpha*sinh(alpha*0)+B*alpha*cosh(alpha*0)", "A*cosh(alpha*x)", "A*alpha*sinh(alpha*L)", "0", "0", "A*cos(alpha*x)+B*sin(alpha*x)", "B*alpha*cos(alpha*0)-A*alpha*sin(alpha*0)", "A*cos(alpha*x)", "-A*alpha*sin(alpha*L)", "(n*pi/L)^2", "T", "C*cos(n*pi*a*t/L)+D*sin(n*pi*a*t/L)", "cos(n*pi*x/L)*[C*cos(n*pi*a*t/L)+D*sin(n*pi*a*t/L)]", "cos(n*pi*a*t/L)", "sin(n*pi*a*t/L)", "cos(n*pi*x/L)", "L", "1", "x", "L/2", "L", "2", "x*cos(n*pi*x/L)", "2*L*[-1+cos(n*pi)]/(n^2*pi^2)", "n*pi*a", "2", "2*cos(n*pi*x/L)", "0", "L/2", "(2*L*[-1+cos(n*pi)]/(n^2*pi^2)*cos(n*pi*a*t/L)+0*sin(n*pi*a*t/L))*cos(n*pi*x/L)" ], "answer_type_v1": [ "EX", "EX", "EX", "TF", "EX", "NV", "NV", "EX", "NV", "TF", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "NV", "NV", "EX", "EX", "EX", "EX", "EX", "TF", "EX", "EX", "EX", "EX", "EX", "EX", "NV", "EX", "EX", "EX", "NV", "EX", "EX", "EX", "NV", "EX", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "For the general solution of the differential equation in X use A and B for your constants and list the functions in alphabetical order, for example $y=A\\cos (x)+B \\sin(x)$. For the differential equation in T use the C and D.For the variable $\\lambda$ type the word lambda and type alpha for $\\alpha$,otherwise treat them as you would any other variable.\nUse the prime notation for derivatives, so the derivative of $X$ is written as $X^\\prime$. Do NOT use $X^\\prime(x)$\nThe longitudinal displacement u(x,t) of a vibrating elastic bar can be modeled by a wave equation with free-end conditions\n a^2 \\frac{\\partial^2 u}{\\partial x^2} = \\frac{\\partial^2 u}{\\partial t^2} , 0< x < L, t > 0 \\frac{\\partial u}{\\partial x} \\Big\\vert_{x=0}=0, \\frac{\\partial u}{\\partial x} \\Big\\vert_{x=L}=0, t > 0 u(x,0)=x \\frac{\\partial u}{\\partial t} \\Big\\vert_{t=0}=-3,0 0 \\frac{\\partial u}{\\partial x} \\Big\\vert_{x=0}=0, \\frac{\\partial u}{\\partial x} \\Big\\vert_{x=L}=0, t > 0 To add in the initial conditions we will have to use a series solution. For the series we relabel the coefficients so that $C=A_n$ and $D=B_n$ $u(x,t)=\\sum\\limits_{n=0}^\\infty \\Bigg(A_n$ [ANS] $+B_n$ [ANS] $\\Bigg)$ [ANS]\nUsing the initial conditions we can compute\n$\\begin{array}{ccccccccccccc}\\hline A_0 & &=& & \\frac{[ANS]}{[ANS]} & & \\int & & L 0 & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccc}\\hline & &=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccccc}\\hline A_n & &=& & \\frac{[ANS]}{[ANS]} & & \\int & & L 0 & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccc}\\hline & &=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccccc}\\hline B_n & &=& & \\frac{[ANS]}{[ANS]} & & \\int & & L 0 & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccc}\\hline & &=& & [ANS] \\\\ \\hline \\end{array}$\nPART 5: Finally!\n$u(x,t)=$ [ANS]+$\\sum\\limits_{n=1}^\\infty$ [ANS]", "answer_v2": [ "a^2*X", "X*T", "X", "T", "X", "0", "0", "L", "0", "T", "A+B*x", "B", "B", "A*cosh(alpha*x)+B*sinh(alpha*x)", "A*alpha*sinh(alpha*0)+B*alpha*cosh(alpha*0)", "A*cosh(alpha*x)", "A*alpha*sinh(alpha*L)", "0", "0", "A*cos(alpha*x)+B*sin(alpha*x)", "B*alpha*cos(alpha*0)-A*alpha*sin(alpha*0)", "A*cos(alpha*x)", "-A*alpha*sin(alpha*L)", "(n*pi/L)^2", "T", "C*cos(n*pi*a*t/L)+D*sin(n*pi*a*t/L)", "cos(n*pi*x/L)*[C*cos(n*pi*a*t/L)+D*sin(n*pi*a*t/L)]", "cos(n*pi*a*t/L)", "sin(n*pi*a*t/L)", "cos(n*pi*x/L)", "L", "1", "x", "L/2", "L", "2", "x*cos(n*pi*x/L)", "2*L*[-1+cos(n*pi)]/(n^2*pi^2)", "n*pi*a", "2", "-3*cos(n*pi*x/L)", "0", "L/2", "(2*L*[-1+cos(n*pi)]/(n^2*pi^2)*cos(n*pi*a*t/L)+0*sin(n*pi*a*t/L))*cos(n*pi*x/L)" ], "answer_type_v2": [ "EX", "EX", "EX", "TF", "EX", "NV", "NV", "EX", "NV", "TF", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "NV", "NV", "EX", "EX", "EX", "EX", "EX", "TF", "EX", "EX", "EX", "EX", "EX", "EX", "NV", "EX", "EX", "EX", "NV", "EX", "EX", "EX", "NV", "EX", "NV", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "For the general solution of the differential equation in X use A and B for your constants and list the functions in alphabetical order, for example $y=A\\cos (x)+B \\sin(x)$. For the differential equation in T use the C and D.For the variable $\\lambda$ type the word lambda and type alpha for $\\alpha$,otherwise treat them as you would any other variable.\nUse the prime notation for derivatives, so the derivative of $X$ is written as $X^\\prime$. Do NOT use $X^\\prime(x)$\nThe longitudinal displacement u(x,t) of a vibrating elastic bar can be modeled by a wave equation with free-end conditions\n a^2 \\frac{\\partial^2 u}{\\partial x^2} = \\frac{\\partial^2 u}{\\partial t^2} , 0< x < L, t > 0 \\frac{\\partial u}{\\partial x} \\Big\\vert_{x=0}=0, \\frac{\\partial u}{\\partial x} \\Big\\vert_{x=L}=0, t > 0 u(x,0)=x \\frac{\\partial u}{\\partial t} \\Big\\vert_{t=0}=-1,0 0 \\frac{\\partial u}{\\partial x} \\Big\\vert_{x=0}=0, \\frac{\\partial u}{\\partial x} \\Big\\vert_{x=L}=0, t > 0 To add in the initial conditions we will have to use a series solution. For the series we relabel the coefficients so that $C=A_n$ and $D=B_n$ $u(x,t)=\\sum\\limits_{n=0}^\\infty \\Bigg(A_n$ [ANS] $+B_n$ [ANS] $\\Bigg)$ [ANS]\nUsing the initial conditions we can compute\n$\\begin{array}{ccccccccccccc}\\hline A_0 & &=& & \\frac{[ANS]}{[ANS]} & & \\int & & L 0 & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccc}\\hline & &=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccccc}\\hline A_n & &=& & \\frac{[ANS]}{[ANS]} & & \\int & & L 0 & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccc}\\hline & &=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccccc}\\hline B_n & &=& & \\frac{[ANS]}{[ANS]} & & \\int & & L 0 & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccc}\\hline & &=& & [ANS] \\\\ \\hline \\end{array}$\nPART 5: Finally!\n$u(x,t)=$ [ANS]+$\\sum\\limits_{n=1}^\\infty$ [ANS]", "answer_v3": [ "a^2*X", "X*T", "X", "T", "X", "0", "0", "L", "0", "T", "A+B*x", "B", "B", "A*cosh(alpha*x)+B*sinh(alpha*x)", "A*alpha*sinh(alpha*0)+B*alpha*cosh(alpha*0)", "A*cosh(alpha*x)", "A*alpha*sinh(alpha*L)", "0", "0", "A*cos(alpha*x)+B*sin(alpha*x)", "B*alpha*cos(alpha*0)-A*alpha*sin(alpha*0)", "A*cos(alpha*x)", "-A*alpha*sin(alpha*L)", "(n*pi/L)^2", "T", "C*cos(n*pi*a*t/L)+D*sin(n*pi*a*t/L)", "cos(n*pi*x/L)*[C*cos(n*pi*a*t/L)+D*sin(n*pi*a*t/L)]", "cos(n*pi*a*t/L)", "sin(n*pi*a*t/L)", "cos(n*pi*x/L)", "L", "1", "x", "L/2", "L", "2", "x*cos(n*pi*x/L)", "2*L*[-1+cos(n*pi)]/(n^2*pi^2)", "n*pi*a", "2", "-1*cos(n*pi*x/L)", "0", "L/2", "(2*L*[-1+cos(n*pi)]/(n^2*pi^2)*cos(n*pi*a*t/L)+0*sin(n*pi*a*t/L))*cos(n*pi*x/L)" ], "answer_type_v3": [ "EX", "EX", "EX", "TF", "EX", "NV", "NV", "EX", "NV", "TF", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "NV", "NV", "EX", "EX", "EX", "EX", "EX", "TF", "EX", "EX", "EX", "EX", "EX", "EX", "NV", "EX", "EX", "EX", "NV", "EX", "EX", "EX", "NV", "EX", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Differential_equations_0319", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Wave equation", "level": "3", "keywords": [ "PDE", "laplace", "equation" ], "problem_v1": "The twist angle $\\theta\\left(x,t\\right)$ of a torsionally vibrating shaft of unit length is determined from: a^2 \\frac{\\partial^2 \\theta}{\\partial x^2} = \\frac{\\partial^2 \\theta}{\\partial t^2} , 0< x < 1, t>0 \\theta\\left(0,t\\right)=0, \\frac{\\partial \\theta}{\\partial x} \\Bigg\\vert_{x=1}=0, t>0 \\theta\\left(x,0\\right)=3x^{2}+x+1, \\frac{\\partial \\theta}{\\partial t} \\Bigg\\vert_{t=0}=0, t>0 Then $\\theta\\left(x,t\\right)=\\sum\\limits_{n=1}^\\infty A_n\\cos\\!\\left(a \\frac{2n-1}{2} \\pi t\\right)\\sin\\!\\left( \\frac{2n-1}{2} \\pi x\\right)$ where $\\begin{array}{ccc}\\hline A_n=& & \\frac{\\int\\limits_0^1 [ANS]dx}{\\int\\limits_0^1 [ANS]dx} \\\\ \\hline \\end{array}$\n$\\begin{array}{ccc}\\hline =& & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "(3*x^2+x+1)*sin((2*n-1)/2*pi*x)", "sin((2*n-1)/2*pi*x)*sin((2*n-1)/2*pi*x)", "-[4*(3*[4*pi*(-1)^n*(2*n-1)+8]+pi*(2*n-1)*[2*(-1)^n+pi*1*(1-2*n)])/[pi^3*(2*n-1)^3]]" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "The twist angle $\\theta\\left(x,t\\right)$ of a torsionally vibrating shaft of unit length is determined from: a^2 \\frac{\\partial^2 \\theta}{\\partial x^2} = \\frac{\\partial^2 \\theta}{\\partial t^2} , 0< x < 1, t>0 \\theta\\left(0,t\\right)=0, \\frac{\\partial \\theta}{\\partial x} \\Bigg\\vert_{x=1}=0, t>0 \\theta\\left(x,0\\right)=-5x^{2}+5x-4, \\frac{\\partial \\theta}{\\partial t} \\Bigg\\vert_{t=0}=0, t>0 Then $\\theta\\left(x,t\\right)=\\sum\\limits_{n=1}^\\infty A_n\\cos\\!\\left(a \\frac{2n-1}{2} \\pi t\\right)\\sin\\!\\left( \\frac{2n-1}{2} \\pi x\\right)$ where $\\begin{array}{ccc}\\hline A_n=& & \\frac{\\int\\limits_0^1 [ANS]dx}{\\int\\limits_0^1 [ANS]dx} \\\\ \\hline \\end{array}$\n$\\begin{array}{ccc}\\hline =& & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "(-5*x^2+5*x-4)*sin((2*n-1)/2*pi*x)", "sin((2*n-1)/2*pi*x)*sin((2*n-1)/2*pi*x)", "-[4*(-5*[4*pi*(-1)^n*(2*n-1)+8]+pi*(2*n-1)*[10*(-1)^n+pi*(-4)*(1-2*n)])/[pi^3*(2*n-1)^3]]" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "The twist angle $\\theta\\left(x,t\\right)$ of a torsionally vibrating shaft of unit length is determined from: a^2 \\frac{\\partial^2 \\theta}{\\partial x^2} = \\frac{\\partial^2 \\theta}{\\partial t^2} , 0< x < 1, t>0 \\theta\\left(0,t\\right)=0, \\frac{\\partial \\theta}{\\partial x} \\Bigg\\vert_{x=1}=0, t>0 \\theta\\left(x,0\\right)=-2x^{2}+x-2, \\frac{\\partial \\theta}{\\partial t} \\Bigg\\vert_{t=0}=0, t>0 Then $\\theta\\left(x,t\\right)=\\sum\\limits_{n=1}^\\infty A_n\\cos\\!\\left(a \\frac{2n-1}{2} \\pi t\\right)\\sin\\!\\left( \\frac{2n-1}{2} \\pi x\\right)$ where $\\begin{array}{ccc}\\hline A_n=& & \\frac{\\int\\limits_0^1 [ANS]dx}{\\int\\limits_0^1 [ANS]dx} \\\\ \\hline \\end{array}$\n$\\begin{array}{ccc}\\hline =& & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "(-2*x^2+x-2)*sin((2*n-1)/2*pi*x)", "sin((2*n-1)/2*pi*x)*sin((2*n-1)/2*pi*x)", "-[4*(-2*[4*pi*(-1)^n*(2*n-1)+8]+pi*(2*n-1)*[2*(-1)^n+pi*(-2)*(1-2*n)])/[pi^3*(2*n-1)^3]]" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Differential_equations_0320", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Laplace's equation", "level": "5", "keywords": [ "PDE", "partial", "differential", "equation", "Laplace equation" ], "problem_v1": "Suppose $u$ is a solution of $\\nabla^2 u=0$ inside the unit circle for the boundary condition that $u$ is equal to 8 for a quarter of the circle, and 6 for the remaining 3 quarters of the circle. Find $u$ at the origin:\n$u(0,0)={}$ [ANS]\nUse what you know about the Poisson formula and your intuition.", "answer_v1": [ "1/4*8+3/4*6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose $u$ is a solution of $\\nabla^2 u=0$ inside the unit circle for the boundary condition that $u$ is equal to 2 for a quarter of the circle, and 9 for the remaining 3 quarters of the circle. Find $u$ at the origin:\n$u(0,0)={}$ [ANS]\nUse what you know about the Poisson formula and your intuition.", "answer_v2": [ "1/4*2+3/4*9" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose $u$ is a solution of $\\nabla^2 u=0$ inside the unit circle for the boundary condition that $u$ is equal to 4 for a quarter of the circle, and 6 for the remaining 3 quarters of the circle. Find $u$ at the origin:\n$u(0,0)={}$ [ANS]\nUse what you know about the Poisson formula and your intuition.", "answer_v3": [ "1/4*4+3/4*6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0321", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Laplace's equation", "level": "4", "keywords": [ "PDE", "partial", "differential", "equation", "Laplace equation" ], "problem_v1": "Solve the Dirichlet problem in the circle of radius 8 using polar coordinates: PDE: $\\quad \\nabla^2 u=u_{rr}+ \\frac{1}{r} u_r+ \\frac{1}{r^2} u_{\\theta\\theta}=0$ for $0 < r < 8$. BC: $\\quad u(8,\\theta)=6 \\sin (7 \\theta)$\n$u(r,\\theta)={}$ [ANS]\n(Write theta for $\\theta$)", "answer_v1": [ "6/(8^7)*r^7*sin(7*theta)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Solve the Dirichlet problem in the circle of radius 2 using polar coordinates: PDE: $\\quad \\nabla^2 u=u_{rr}+ \\frac{1}{r} u_r+ \\frac{1}{r^2} u_{\\theta\\theta}=0$ for $0 < r < 2$. BC: $\\quad u(2,\\theta)=9 \\sin (3 \\theta)$\n$u(r,\\theta)={}$ [ANS]\n(Write theta for $\\theta$)", "answer_v2": [ "9/(2^3)*r^3*sin(3*theta)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Solve the Dirichlet problem in the circle of radius 4 using polar coordinates: PDE: $\\quad \\nabla^2 u=u_{rr}+ \\frac{1}{r} u_r+ \\frac{1}{r^2} u_{\\theta\\theta}=0$ for $0 < r < 4$. BC: $\\quad u(4,\\theta)=6 \\sin (3 \\theta)$\n$u(r,\\theta)={}$ [ANS]\n(Write theta for $\\theta$)", "answer_v3": [ "6/(4^3)*r^3*sin(3*theta)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Differential_equations_0322", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Other separable equations", "level": "5", "keywords": [ "partial", "differential", "equation", "separable" ], "problem_v1": "$\\lambda$ is typed as lambda, $\\alpha$ as alpha.\nThe PDE k^2 \\frac{\\partial^2 u}{\\partial x^2} =y^8 \\frac{\\partial u}{\\partial y} is separable, so we look for solutions of the form $u(x,t)=X(x)Y(y)$. When solving DE in X and Y use the constants a and b for X and c for Y. The PDE can be rewritten using this solution as (placing constants in the DE for Y) into [ANS]=[ANS]=$-\\lambda$\nNote: Use the prime notation for derivatives, so the derivative of $X$ is written as $X^\\prime$. Do NOT use $X^\\prime(x)$\nSince these differential equations are independent of each other, they can be separated DE in X: [ANS] $=0$ DE in Y: [ANS] $=0$\nNow we solve the separate separated ODEs for the different cases in $\\lambda$. In each case the general solution in X is written with constants a and b and the general solution in Y is written with constants c and d. Write the functions alphabetically, so that if the solutions involve cos and sin, your answer would be acos(x)+bsin(x). Case 1: $\\lambda=0$ $X(x)=$ [ANS]\n$Y(y)=$ [ANS]\nDE in Y If $\\lambda\\neq 0$, the differential equation in Y is first order, linear, and more importantly separable. We separate the two sides as [ANS]=[ANS]\nIntegrating both sides with respect to $y$ (placing the constant of integration c in the right hand side) we get [ANS]=[ANS]\nSolving for Y, using the funny algebra of constant where $e^c=c$ is just another constant we get $Y=$ [ANS]\nFor $\\lambda\\neq 0$ we get a Sturm-Louiville problem in X which we need to handle two more cases Case 2: $\\lambda=-\\alpha^2$ $X(x)=$ [ANS]\nCase 3: $\\lambda=\\alpha^2$ $X(x)=$ [ANS]\nFinal Solution Case 1: $\\lambda=0$ $u=$ [ANS]\nCase 2: $\\lambda=-\\alpha^2$ $u=$ [ANS]\nCase 3: $\\lambda=\\alpha^2$ $u=$ [ANS]", "answer_v1": [ "X", "y^8*Y", "X", "y^8*Y", "a+b*x", "c", "Y", "-lambda*k^2/(y^8)", "ln(|Y|)", "-lambda*k^2/(-7)*y^(-7)+C", "c*e^[lambda*k^2/7*y^(-7)]", "a*cosh(alpha*x)+b*sinh(alpha*x)", "a*cos(alpha*x)+b*sin(alpha*x)", "a+b*x", "[a*cosh(alpha*x)+b*sinh(alpha*x)]*e^[-alpha^2*k^2/7*y^(-7)]", "[a*cos(alpha*x)+b*sin(alpha*x)]*e^[alpha^2*k^2/7*y^(-7)]" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX", "EX", "TF", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "$\\lambda$ is typed as lambda, $\\alpha$ as alpha.\nThe PDE k^2 \\frac{\\partial^2 u}{\\partial x^2} =y^2 \\frac{\\partial u}{\\partial y} is separable, so we look for solutions of the form $u(x,t)=X(x)Y(y)$. When solving DE in X and Y use the constants a and b for X and c for Y. The PDE can be rewritten using this solution as (placing constants in the DE for Y) into [ANS]=[ANS]=$-\\lambda$\nNote: Use the prime notation for derivatives, so the derivative of $X$ is written as $X^\\prime$. Do NOT use $X^\\prime(x)$\nSince these differential equations are independent of each other, they can be separated DE in X: [ANS] $=0$ DE in Y: [ANS] $=0$\nNow we solve the separate separated ODEs for the different cases in $\\lambda$. In each case the general solution in X is written with constants a and b and the general solution in Y is written with constants c and d. Write the functions alphabetically, so that if the solutions involve cos and sin, your answer would be acos(x)+bsin(x). Case 1: $\\lambda=0$ $X(x)=$ [ANS]\n$Y(y)=$ [ANS]\nDE in Y If $\\lambda\\neq 0$, the differential equation in Y is first order, linear, and more importantly separable. We separate the two sides as [ANS]=[ANS]\nIntegrating both sides with respect to $y$ (placing the constant of integration c in the right hand side) we get [ANS]=[ANS]\nSolving for Y, using the funny algebra of constant where $e^c=c$ is just another constant we get $Y=$ [ANS]\nFor $\\lambda\\neq 0$ we get a Sturm-Louiville problem in X which we need to handle two more cases Case 2: $\\lambda=-\\alpha^2$ $X(x)=$ [ANS]\nCase 3: $\\lambda=\\alpha^2$ $X(x)=$ [ANS]\nFinal Solution Case 1: $\\lambda=0$ $u=$ [ANS]\nCase 2: $\\lambda=-\\alpha^2$ $u=$ [ANS]\nCase 3: $\\lambda=\\alpha^2$ $u=$ [ANS]", "answer_v2": [ "X", "y^2*Y", "X", "y^2*Y", "a+b*x", "c", "Y", "-lambda*k^2/(y^2)", "ln(|Y|)", "-lambda*k^2/(-1)*y^(-1)+C", "c*e^[lambda*k^2/1*y^(-1)]", "a*cosh(alpha*x)+b*sinh(alpha*x)", "a*cos(alpha*x)+b*sin(alpha*x)", "a+b*x", "[a*cosh(alpha*x)+b*sinh(alpha*x)]*e^[-alpha^2*k^2/1*y^(-1)]", "[a*cos(alpha*x)+b*sin(alpha*x)]*e^[alpha^2*k^2/1*y^(-1)]" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX", "EX", "TF", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "$\\lambda$ is typed as lambda, $\\alpha$ as alpha.\nThe PDE k^2 \\frac{\\partial^2 u}{\\partial x^2} =y^4 \\frac{\\partial u}{\\partial y} is separable, so we look for solutions of the form $u(x,t)=X(x)Y(y)$. When solving DE in X and Y use the constants a and b for X and c for Y. The PDE can be rewritten using this solution as (placing constants in the DE for Y) into [ANS]=[ANS]=$-\\lambda$\nNote: Use the prime notation for derivatives, so the derivative of $X$ is written as $X^\\prime$. Do NOT use $X^\\prime(x)$\nSince these differential equations are independent of each other, they can be separated DE in X: [ANS] $=0$ DE in Y: [ANS] $=0$\nNow we solve the separate separated ODEs for the different cases in $\\lambda$. In each case the general solution in X is written with constants a and b and the general solution in Y is written with constants c and d. Write the functions alphabetically, so that if the solutions involve cos and sin, your answer would be acos(x)+bsin(x). Case 1: $\\lambda=0$ $X(x)=$ [ANS]\n$Y(y)=$ [ANS]\nDE in Y If $\\lambda\\neq 0$, the differential equation in Y is first order, linear, and more importantly separable. We separate the two sides as [ANS]=[ANS]\nIntegrating both sides with respect to $y$ (placing the constant of integration c in the right hand side) we get [ANS]=[ANS]\nSolving for Y, using the funny algebra of constant where $e^c=c$ is just another constant we get $Y=$ [ANS]\nFor $\\lambda\\neq 0$ we get a Sturm-Louiville problem in X which we need to handle two more cases Case 2: $\\lambda=-\\alpha^2$ $X(x)=$ [ANS]\nCase 3: $\\lambda=\\alpha^2$ $X(x)=$ [ANS]\nFinal Solution Case 1: $\\lambda=0$ $u=$ [ANS]\nCase 2: $\\lambda=-\\alpha^2$ $u=$ [ANS]\nCase 3: $\\lambda=\\alpha^2$ $u=$ [ANS]", "answer_v3": [ "X", "y^4*Y", "X", "y^4*Y", "a+b*x", "c", "Y", "-lambda*k^2/(y^4)", "ln(|Y|)", "-lambda*k^2/(-3)*y^(-3)+C", "c*e^[lambda*k^2/3*y^(-3)]", "a*cosh(alpha*x)+b*sinh(alpha*x)", "a*cos(alpha*x)+b*sin(alpha*x)", "a+b*x", "[a*cosh(alpha*x)+b*sinh(alpha*x)]*e^[-alpha^2*k^2/3*y^(-3)]", "[a*cos(alpha*x)+b*sin(alpha*x)]*e^[alpha^2*k^2/3*y^(-3)]" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX", "EX", "TF", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Differential_equations_0323", "subject": "Differential_equations", "topic": "Partial differential equations", "subtopic": "Inhomogenous equations", "level": "5", "keywords": [ "PDE", "non-homogeneous", "boundary", "value" ], "problem_v1": "In this problem we explore using Fourier series to solve nonhomogeneous boundary value problems. For $u_n$ type un, for derivatives use the prime notation $u_n^\\prime,u_n^{\\prime\\prime},\\ldots$.\nSolve the heat equation\n$ \\frac{\\partial^2 u}{\\partial x^2} +xe^{-7t}= \\frac{\\partial u}{\\partial t} , 0< x < 5, t > 0$\n$ u(0,t)=0, u(5,t)=0, t > 0$\n$ u(x,0)=x, 0 < x < 5$\nTo find a series solution u we first must write the function $xe^{-7t}$ as a Fourier series $xe^{-7t}=\\sum\\limits_{n=1}^\\infty F_n\\sin\\left( \\frac{n\\pi}{5} x\\right)$ Therefore\n$ F_n= \\frac{2}{5} \\int_0^{5}$ [ANS] $ dx$\n$=$ [ANS]\nNow we try to find a solution $u$ of the form $u(x,t)=\\sum\\limits_{n=1}^\\infty u_n(t)\\sin\\left( \\frac{n\\pi}{5} x\\right)$ Using this series in the PDE we get\n$ \\frac{\\partial u}{\\partial t} - \\frac{\\partial^2 u}{\\partial x^2} $\n$=\\sum\\limits_{n=1}^\\infty$ [ANS] $\\sin\\left( \\frac{n\\pi}{5} x\\right)$\nSince we want $ \\frac{\\partial u}{\\partial t} - \\frac{\\partial^2 u}{\\partial x^2} =xe^{-7t}$ their Fourier coefficients must be equal: [ANS] $=F_n$ which gives us an ODE in $u_n$ which we solve using constant $c_n$, to get $u_n(t)=$ [ANS]\nNow that we have a general form for u, we can find the constants $c_n$ by using the initial condition $u(x,0)=x$. Plugging the formula we just derived for $u_n(t)$ into the series for u we get $u(x,0)=\\sum\\limits_{n=1}^\\infty$ [ANS] $\\sin\\left( \\frac{n\\pi}{5} x\\right)=x$ Recognizing that this is a Fourier series for $x$, we can solve for $c_n$:\n$ c_n= \\frac{2}{5} \\int_0^{5}$ [ANS] $ dx-$ [ANS]\n$=$ [ANS]", "answer_v1": [ "x*e^(-7*t)*sin(n*pi*x/5)", "2*e^(-7*t)*5*(-1)^{n+1}/(n*pi)", "(n*pi/5)^2*un+un", "(n*pi/5)^2*un+un", "125*2*(-1)^n/[e^(7*t)*n*pi*(175-n^2*pi^2)]+cn/[e^(n^2*pi^2*t/25)]", "125*2*(-1)^n/[n*pi*(175-n^2*pi^2)]+cn", "x*sin(n*pi*x/5)", "2*(-1)^n*125/[n*pi*(175-n^2*pi^2)]", "2*(-1)^{n+1}*5/(n*pi)-2*(-1)^n*125/[n*pi*(175-n^2*pi^2)]" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "In this problem we explore using Fourier series to solve nonhomogeneous boundary value problems. For $u_n$ type un, for derivatives use the prime notation $u_n^\\prime,u_n^{\\prime\\prime},\\ldots$.\nSolve the heat equation\n$ \\frac{\\partial^2 u}{\\partial x^2} +xe^{-2t}= \\frac{\\partial u}{\\partial t} , 0< x < 7, t > 0$\n$ u(0,t)=0, u(7,t)=0, t > 0$\n$ u(x,0)=x, 0 < x < 7$\nTo find a series solution u we first must write the function $xe^{-2t}$ as a Fourier series $xe^{-2t}=\\sum\\limits_{n=1}^\\infty F_n\\sin\\left( \\frac{n\\pi}{7} x\\right)$ Therefore\n$ F_n= \\frac{2}{7} \\int_0^{7}$ [ANS] $ dx$\n$=$ [ANS]\nNow we try to find a solution $u$ of the form $u(x,t)=\\sum\\limits_{n=1}^\\infty u_n(t)\\sin\\left( \\frac{n\\pi}{7} x\\right)$ Using this series in the PDE we get\n$ \\frac{\\partial u}{\\partial t} - \\frac{\\partial^2 u}{\\partial x^2} $\n$=\\sum\\limits_{n=1}^\\infty$ [ANS] $\\sin\\left( \\frac{n\\pi}{7} x\\right)$\nSince we want $ \\frac{\\partial u}{\\partial t} - \\frac{\\partial^2 u}{\\partial x^2} =xe^{-2t}$ their Fourier coefficients must be equal: [ANS] $=F_n$ which gives us an ODE in $u_n$ which we solve using constant $c_n$, to get $u_n(t)=$ [ANS]\nNow that we have a general form for u, we can find the constants $c_n$ by using the initial condition $u(x,0)=x$. Plugging the formula we just derived for $u_n(t)$ into the series for u we get $u(x,0)=\\sum\\limits_{n=1}^\\infty$ [ANS] $\\sin\\left( \\frac{n\\pi}{7} x\\right)=x$ Recognizing that this is a Fourier series for $x$, we can solve for $c_n$:\n$ c_n= \\frac{2}{7} \\int_0^{7}$ [ANS] $ dx-$ [ANS]\n$=$ [ANS]", "answer_v2": [ "x*e^(-2*t)*sin(n*pi*x/7)", "2*e^(-2*t)*7*(-1)^{n+1}/(n*pi)", "(n*pi/7)^2*un+un", "(n*pi/7)^2*un+un", "343*2*(-1)^n/[e^(2*t)*n*pi*(98-n^2*pi^2)]+cn/[e^(n^2*pi^2*t/49)]", "343*2*(-1)^n/[n*pi*(98-n^2*pi^2)]+cn", "x*sin(n*pi*x/7)", "2*(-1)^n*343/[n*pi*(98-n^2*pi^2)]", "2*(-1)^{n+1}*7/(n*pi)-2*(-1)^n*343/[n*pi*(98-n^2*pi^2)]" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "In this problem we explore using Fourier series to solve nonhomogeneous boundary value problems. For $u_n$ type un, for derivatives use the prime notation $u_n^\\prime,u_n^{\\prime\\prime},\\ldots$.\nSolve the heat equation\n$ \\frac{\\partial^2 u}{\\partial x^2} +xe^{-4t}= \\frac{\\partial u}{\\partial t} , 0< x < 5, t > 0$\n$ u(0,t)=0, u(5,t)=0, t > 0$\n$ u(x,0)=x, 0 < x < 5$\nTo find a series solution u we first must write the function $xe^{-4t}$ as a Fourier series $xe^{-4t}=\\sum\\limits_{n=1}^\\infty F_n\\sin\\left( \\frac{n\\pi}{5} x\\right)$ Therefore\n$ F_n= \\frac{2}{5} \\int_0^{5}$ [ANS] $ dx$\n$=$ [ANS]\nNow we try to find a solution $u$ of the form $u(x,t)=\\sum\\limits_{n=1}^\\infty u_n(t)\\sin\\left( \\frac{n\\pi}{5} x\\right)$ Using this series in the PDE we get\n$ \\frac{\\partial u}{\\partial t} - \\frac{\\partial^2 u}{\\partial x^2} $\n$=\\sum\\limits_{n=1}^\\infty$ [ANS] $\\sin\\left( \\frac{n\\pi}{5} x\\right)$\nSince we want $ \\frac{\\partial u}{\\partial t} - \\frac{\\partial^2 u}{\\partial x^2} =xe^{-4t}$ their Fourier coefficients must be equal: [ANS] $=F_n$ which gives us an ODE in $u_n$ which we solve using constant $c_n$, to get $u_n(t)=$ [ANS]\nNow that we have a general form for u, we can find the constants $c_n$ by using the initial condition $u(x,0)=x$. Plugging the formula we just derived for $u_n(t)$ into the series for u we get $u(x,0)=\\sum\\limits_{n=1}^\\infty$ [ANS] $\\sin\\left( \\frac{n\\pi}{5} x\\right)=x$ Recognizing that this is a Fourier series for $x$, we can solve for $c_n$:\n$ c_n= \\frac{2}{5} \\int_0^{5}$ [ANS] $ dx-$ [ANS]\n$=$ [ANS]", "answer_v3": [ "x*e^{-4*t}*sin(n*pi*x/5)", "2*e^{-4*t}*5*(-1)^{n+1}/(n*pi)", "(n*pi/5)^2*un+un", "(n*pi/5)^2*un+un", "125*2*(-1)^n/[e^(4*t)*n*pi*(100-n^2*pi^2)]+cn/[e^(n^2*pi^2*t/25)]", "125*2*(-1)^n/[n*pi*(100-n^2*pi^2)]+cn", "x*sin(n*pi*x/5)", "2*(-1)^n*125/[n*pi*(100-n^2*pi^2)]", "2*(-1)^{n+1}*5/(n*pi)-2*(-1)^n*125/[n*pi*(100-n^2*pi^2)]" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] } ]