Question 1

Which of the following is the general solution of the homogeneous second-order differential equation   
are arbitrary real constants.

Accepted Answer

A)  $ y=C\left(\cos \frac{t}{3}+\sin \frac{t}{3}\right) $ 
B)  $ y=C_{1} \cos (3 t)+C_{2} \sin (3 t) $ 
C)  $ y=C(\cos (3 t)+\sin (3 t)) $ 
D)  $ y=C_{1} \cos \left(\frac{t}{3}\right)+C_{2} \sin \left(\frac{t}{3}\right) $ 
E)  $ y=\cos \left(\frac{t}{3}\right)+\sin \left(\frac{t}{3}\right)+C t $
F) $ y=\cos (3 t)+\sin (3 t)+C $ 
A)  $ y=C\left(\cos \frac{t}{3}+\sin \frac{t}{3}\right) $ 
B)  $ y=C_{1} \cos (3 t)+C_{2} \sin (3 t) $ 
C)  $ y=C(\cos (3 t)+\sin (3 t)) $ 
D)  $ y=C_{1} \cos \left(\frac{t}{3}\right)+C_{2} \sin \left(\frac{t}{3}\right) $ 
E)  $ y=\cos \left(\frac{t}{3}\right)+\sin \left(\frac{t}{3}\right)+C t $
F) $ y=\cos (3 t)+\sin (3 t)+C $

Question 2

If Y1 and Y2 are both solutions of the differential equation   then Y1 - Y2 is also a solution of this equation.

Accepted Answer

A) True 
 B)False

Question 3

For what value(s) of $\alpha$ is y =   a solution of the second-order homogeneous differential equation

Accepted Answer

A)    
B)  0 and   
C)  0 and -   
D)  -   
E)  -   and   
A)    
B)  0 and   
C)  0 and -   
D)  -   
E)  -   and

Question 4

Consider this initial value problem:
 
Which of the following is an accurate description of the long-term behavior of the solution?

Accepted Answer

A)  y(t) tends to 0 as t $\rightarrow$ $\infty$. 
B)  y(t) is strictly increasing and approaches $\infty$ as t $\rightarrow$ $\infty$. 
C)  y(t) is strictly decreasing and approaches -$\infty$ as t $\rightarrow$ $\infty$. 
D)  y(t) becomes unbounded in both the positive and negative y-direction as t $\rightarrow$ $\infty$. 
A)  y(t) tends to 0 as t $\rightarrow$ $\infty$. 
B)  y(t) is strictly increasing and approaches $\infty$ as t $\rightarrow$ $\infty$. 
C)  y(t) is strictly decreasing and approaches -$\infty$ as t $\rightarrow$ $\infty$. 
D)  y(t) becomes unbounded in both the positive and negative y-direction as t $\rightarrow$ $\infty$.

Question 5

Suppose a 64-lb object stretches a spring 2 feet while in equilibrium, and a dashpot provides a damping force of 3 lbs for every foot per second of velocity. The form of the equation of unforced motion of the object in such a spring-mass system is   where m is the mass of the object, c is the damping constant, and k is the spring constant. Assume that the spring starts at a height of 5/2 feet below the horizontal with an upward velocity of - 3/2 feet per second.
For what values of $\alpha$ and $\beta$is the function   the general solution of the equation of motion for this spring-mass system? Provide exact values, not decimal approximations.
$\alpha$ = _______________,$\beta$ = _______________

Accepted Answer

The answer of Suppose a 64-lb object stretches a spring...

Question 6

Consider the nonhomogeneous differential equation   What is the general solution of this differential equation?

Accepted Answer

The answer of Consider the nonhomogeneous differential equation  ...

Question 7

Consider this second-order nonhomogeneous differential equation:
 
Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C1 and C2 are arbitrary real constants.

Accepted Answer

A)  $ y(t)=C_{1} \sin (5 t)+C_{2} \cos (5 t) $ 
B)  $ y(t)=C_{1} \sin (25 t)+C_{2} \cos (25 t) $ 
C)  $ y(t)=C_{1}+C_{2} e^{-5 t} $ 
D)  $ y(t)=C_{1} e^{-5 t}+C_{2} e^{5 t} $ 
A)  $ y(t)=C_{1} \sin (5 t)+C_{2} \cos (5 t) $ 
B)  $ y(t)=C_{1} \sin (25 t)+C_{2} \cos (25 t) $ 
C)  $ y(t)=C_{1}+C_{2} e^{-5 t} $ 
D)  $ y(t)=C_{1} e^{-5 t}+C_{2} e^{5 t} $

Question 8

Use variation of parameters to find the general solution of the nonhomogeneous differential equation

Accepted Answer

A)  $ y(t)=e^{2 t}\left(C_{1}+C_{2} t\right)+\frac{1}{16} e^{2 t}(\cos (4 t)-\sin (4 t)) $ 
B)  $ y(t)=e^{2 t}\left(C_{1}+C_{2} t\right)-\frac{1}{16} e^{2 t} \sin (4 t) $ 
C)  $ y(t)=e^{-2 t}\left(C_{1}+C_{2} t\right)-\frac{1}{16} e^{-2 t} \sin (4 t) $ 
D)  $ \left.y(t)=e^{-2 t}\left(C_{1}+C_{2} t\right)+\frac{1}{16} e^{-2 t} \cdot \cos (4 t)-\sin (4 t)\right) $ 
A)  $ y(t)=e^{2 t}\left(C_{1}+C_{2} t\right)+\frac{1}{16} e^{2 t}(\cos (4 t)-\sin (4 t)) $ 
B)  $ y(t)=e^{2 t}\left(C_{1}+C_{2} t\right)-\frac{1}{16} e^{2 t} \sin (4 t) $ 
C)  $ y(t)=e^{-2 t}\left(C_{1}+C_{2} t\right)-\frac{1}{16} e^{-2 t} \sin (4 t) $ 
D)  $ \left.y(t)=e^{-2 t}\left(C_{1}+C_{2} t\right)+\frac{1}{16} e^{-2 t} \cdot \cos (4 t)-\sin (4 t)\right) $

Question 9

Consider this second-order nonhomogeneous differential equation:
 
Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C1 and C2 are arbitrary real constants.

Accepted Answer

A)  $ y(t)=C_{1} e^{-3 t}+C_{2} e^{7 t} $ 
B)  $ y(t)=C_{1} e^{-3 t}+C_{2} t e^{-3 t} $ 
C)  $ y(t)=C_{1} e^{3 t}+C_{2} t e^{3 t} $ 
D)  $ y(t)=C_{1} e^{3 t}+C_{2} e^{-7 t} $ 
E)  $ y(t)=C_{1} e^{-7 t}+C_{2} t e^{-7 t} $ 
A)  $ y(t)=C_{1} e^{-3 t}+C_{2} e^{7 t} $ 
B)  $ y(t)=C_{1} e^{-3 t}+C_{2} t e^{-3 t} $ 
C)  $ y(t)=C_{1} e^{3 t}+C_{2} t e^{3 t} $ 
D)  $ y(t)=C_{1} e^{3 t}+C_{2} e^{-7 t} $ 
E)  $ y(t)=C_{1} e^{-7 t}+C_{2} t e^{-7 t} $

Question 10

Which of these is the general solution of the second-order nonhomogeneous differential equation   
and all capital letters are arbitrary real constants.

Accepted Answer

A)  $ y(t)=C_{1} e^{\frac{1}{2} t}+C_{2} t e^{\frac{1}{2} t}+\left(A t^{2}+B t+C\right) e^{\frac{1}{2} t}+D $ 
B)  $ y(t)=C_{1} e^{\frac{1}{2} t}+C_{2} t e^{\frac{1}{2} t}+A e^{\frac{1}{2} t}+B $ 
C)  $ y(t)=e^{\frac{1}{2} t}\left(C_{1}+C_{2} t\right)+(A t+
B)  e^{\frac{1}{2} t}+C $ 
D)  $ y(t)=e^{-\frac{1}{2} t}\left(C_{1}+C_{2} t\right)+\left(A t^{2}+B t+C\right) e^{\frac{1}{2} t}+D $ 
E)  $ y(t)=e^{-\frac{1}{2} t}\left(C_{1}+C_{2} t\right)+A e^{\frac{1}{2} t}+B $ 
A)  $ y(t)=C_{1} e^{\frac{1}{2} t}+C_{2} t e^{\frac{1}{2} t}+\left(A t^{2}+B t+C\right) e^{\frac{1}{2} t}+D $ 
B)  $ y(t)=C_{1} e^{\frac{1}{2} t}+C_{2} t e^{\frac{1}{2} t}+A e^{\frac{1}{2} t}+B $ 
C)  $ y(t)=e^{\frac{1}{2} t}\left(C_{1}+C_{2} t\right)+(A t+
B)  e^{\frac{1}{2} t}+C $ 
D)  $ y(t)=e^{-\frac{1}{2} t}\left(C_{1}+C_{2} t\right)+\left(A t^{2}+B t+C\right) e^{\frac{1}{2} t}+D $ 
E)  $ y(t)=e^{-\frac{1}{2} t}\left(C_{1}+C_{2} t\right)+A e^{\frac{1}{2} t}+B $

Question 11

Suppose a 12-lb object stretches a spring 2 feet while in equilibrium. If the object is displaced an additional 2 inches and is then set in motion with an initial upward velocity of -1 feet per second.
For what values of the arbitrary constants C1 and C2 does the general solution   satisfy the initial conditions? Provide the exact values, not decimal approximations.
C1 = ________, C2 = ________

Accepted Answer

Question 12

Suppose a 10-lb object stretches a spring 2.25 feet while in equilibrium. If the object is displaced an additional 1.5 inches and is then set in motion with an initial upward velocity of -1 feet per second.
Suppose the phase angle δ is such that the solution curve can be expressed in the form .   What is an expression for tan δ? Provide the exact value, not a decimal approximation.
tan δ = ________

Accepted Answer

The answer of Suppose a 10-lb object stretches a spring...

Question 13

Consider the differential equation   
Which of the following statements is true?

Accepted Answer

A)  If 2   is a solution of this differential equation, then so is   . 
B)  If Y1 and Y2 are both solutions of this differential equation, then Y1 - Y2 cannot be a solution of it. 
C)  The Principle of Superposition guarantees that if y1 and y2 are both solutions of this differential equation, then C1 y1 + C2 y2 must also be a solution of it, for any choice of real constants and . 
D)  There exist nonzero real constants C1 and C2 such that C1 y1 - C2 y2 is a solution of this differential equation. 
A)  If 2   is a solution of this differential equation, then so is   . 
B)  If Y1 and Y2 are both solutions of this differential equation, then Y1 - Y2 cannot be a solution of it. 
C)  The Principle of Superposition guarantees that if y1 and y2 are both solutions of this differential equation, then C1 y1 + C2 y2 must also be a solution of it, for any choice of real constants and . 
D)  There exist nonzero real constants C1 and C2 such that C1 y1 - C2 y2 is a solution of this differential equation.

Question 14

Consider this second-order nonhomogeneous differential equation:
 
Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C1 and C2 are arbitrary real constants.

Accepted Answer

A)  $ y(t)=e^{\frac{2}{5} t}\left(C_{1}+C_{2} t\right) $ 
B)  $ y(t)=C_{1} e^{\frac{2}{5} t}+C_{2} e^{-\frac{2}{5} t} $ 
C)  $ y(t)=e^{\frac{2}{5} t}\left(t+C_{1}\right)+C_{2} $ 
D)  $ y(t)=e^{-\frac{2}{5} t}\left(C_{1}+C_{2} t\right) $ 
E)  $ y(t)=e^{-\frac{2}{5} t}\left(t+C_{1}\right)+C_{2} $ 
A)  $ y(t)=e^{\frac{2}{5} t}\left(C_{1}+C_{2} t\right) $ 
B)  $ y(t)=C_{1} e^{\frac{2}{5} t}+C_{2} e^{-\frac{2}{5} t} $ 
C)  $ y(t)=e^{\frac{2}{5} t}\left(t+C_{1}\right)+C_{2} $ 
D)  $ y(t)=e^{-\frac{2}{5} t}\left(C_{1}+C_{2} t\right) $ 
E)  $ y(t)=e^{-\frac{2}{5} t}\left(t+C_{1}\right)+C_{2} $

Question 15

Use the method of reduction of order to find a second solution of the differential equation   using the fact that y1 = t-1 is a solution. is a solution.

Accepted Answer

Question 16

Suppose a 160-lb object stretches a spring 3 feet while in equilibrium, and a dashpot provides a damping force of 3 lbs for every foot per second of velocity. The form of the equation of unforced motion of the object in such a spring-mass system is   where m is the mass of the object, c is the damping constant, and k is the spring constant. Assume that the spring starts at a height of 1/2 feet below the horizontal with an upward velocity of - 3 feet per second.
After how many seconds does the object pass through the equilibrium position for the first time? Round your answer to the nearest hundredth of a second.

Accepted Answer

The answer of Suppose a 160-lb object stretches a spring...

Question 17

Which of these is the general solution of the second-order nonhomogeneous differential equation   , and all capital letters are arbitrary real constants.

Accepted Answer

A)  $ y(t)=C_{1} e^{t}+C_{2} t e^{t}+A \sin (\sqrt{7} t)+B \cos (\sqrt{7} t)+C \sin \left(\frac{5 \pi}{2} t\right)+D \cos \left(\frac{5 \pi}{2} t\right) $ 
B)  $ y(t)=C_{1}+C_{2} t+A \sin (\sqrt{7} t)+B \cos (\sqrt{7} t)+C \sin \left(\frac{5 \pi}{2} t\right)+D \cos \left(\frac{5 \pi}{2} t\right) $ 
C)  $ y(t)=C_{1}+C_{2} t+A \sin (\sqrt{7} t)+B \cos \left(\frac{5 \pi}{2} t\right) $ 
D)  $ y(t)=C_{1} t+A \sin (\sqrt{7} t)+B \cos \left(\frac{5 \pi}{2} t\right) $ 
E)  $ y(t)=C_{1} t+A \sin (\sqrt{7} t)+B \cos (\sqrt{7} t)+C \sin \left(\frac{5 \pi}{2} t\right)+D \cos \left(\frac{5 \pi}{2} t\right) $ 
A)  $ y(t)=C_{1} e^{t}+C_{2} t e^{t}+A \sin (\sqrt{7} t)+B \cos (\sqrt{7} t)+C \sin \left(\frac{5 \pi}{2} t\right)+D \cos \left(\frac{5 \pi}{2} t\right) $ 
B)  $ y(t)=C_{1}+C_{2} t+A \sin (\sqrt{7} t)+B \cos (\sqrt{7} t)+C \sin \left(\frac{5 \pi}{2} t\right)+D \cos \left(\frac{5 \pi}{2} t\right) $ 
C)  $ y(t)=C_{1}+C_{2} t+A \sin (\sqrt{7} t)+B \cos \left(\frac{5 \pi}{2} t\right) $ 
D)  $ y(t)=C_{1} t+A \sin (\sqrt{7} t)+B \cos \left(\frac{5 \pi}{2} t\right) $ 
E)  $ y(t)=C_{1} t+A \sin (\sqrt{7} t)+B \cos (\sqrt{7} t)+C \sin \left(\frac{5 \pi}{2} t\right)+D \cos \left(\frac{5 \pi}{2} t\right) $

Question 18

Suppose a 6-lb object stretches a spring 1.75 feet while in equilibrium. If the object is displaced an additional 1 inches and is then set in motion with an initial upward velocity of -0.8 feet per second.
For what values of $\alpha$ and $\beta$is the function   the general solution of the equation of motion for this spring-mass system? Round your answer to three decimal places.
$\alpha$ = ________, $\beta$ = ________

Accepted Answer

The answer of Suppose a 6-lb object stretches a spring...

Question 19

Use variation of parameters to find the general solution of the nonhomogeneous differential equation

Accepted Answer

A)  $ \left.y(t)=C_{1} e^{t}+C_{1} t e^{t}-\frac{1}{8} e^{t} \ln \left(1+4 t^{2}\right)+\frac{1}{2} e^{t} \tan ^{-1} 2 t\right) $ 
B)  $ \left.y(t)=C_{1} e^{t}+C_{1} t e^{t}-\frac{1}{8} t e^{t} \ln \left(1+4 t^{2}\right)+\frac{1}{2} e^{t} \tan ^{-1} 2 t\right) $ 
C)  $ \left.y(t)=C_{1} e^{t}+C_{1} t e^{t}-8 e^{t} \ln \left(1+4 t^{2}\right)+2 t e^{t} \tan ^{-1} 2 t\right) $ 
D)  $ \left.y(t)=C_{1} e^{t}+C_{1} t e^{t}-8 t e^{t} \ln \left(1+4 t^{2}\right)+2 e^{t} \tan ^{-1} 2 t\right) $ 
A)  $ \left.y(t)=C_{1} e^{t}+C_{1} t e^{t}-\frac{1}{8} e^{t} \ln \left(1+4 t^{2}\right)+\frac{1}{2} e^{t} \tan ^{-1} 2 t\right) $ 
B)  $ \left.y(t)=C_{1} e^{t}+C_{1} t e^{t}-\frac{1}{8} t e^{t} \ln \left(1+4 t^{2}\right)+\frac{1}{2} e^{t} \tan ^{-1} 2 t\right) $ 
C)  $ \left.y(t)=C_{1} e^{t}+C_{1} t e^{t}-8 e^{t} \ln \left(1+4 t^{2}\right)+2 t e^{t} \tan ^{-1} 2 t\right) $ 
D)  $ \left.y(t)=C_{1} e^{t}+C_{1} t e^{t}-8 t e^{t} \ln \left(1+4 t^{2}\right)+2 e^{t} \tan ^{-1} 2 t\right) $

Question 20

Consider this second-order nonhomogeneous differential equation:   Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C1 and C2 are arbitrary real constants.

Accepted Answer

A)  $ y(t)=C_{1} e^{3 t}(\sin (4 t)+\cos (4 t))+C_{2} $ 
B)  $ y(t)=C_{1} e^{4 t}\left(\sin (3 t)+C_{2} e^{4 t}(\cos (3 t)\right. $ 
C)  $ y(t)=C_{1} e^{4 t}(\sin (3 t)+\cos (3 t))+C_{2} $ 
D)  $ y(t)=C_{1} e^{3 t}\left(\sin (4 t)+C_{2} e^{3 t}(\cos (4 t)\right. $ 
A)  $ y(t)=C_{1} e^{3 t}(\sin (4 t)+\cos (4 t))+C_{2} $ 
B)  $ y(t)=C_{1} e^{4 t}\left(\sin (3 t)+C_{2} e^{4 t}(\cos (3 t)\right. $ 
C)  $ y(t)=C_{1} e^{4 t}(\sin (3 t)+\cos (3 t))+C_{2} $ 
D)  $ y(t)=C_{1} e^{3 t}\left(\sin (4 t)+C_{2} e^{3 t}(\cos (4 t)\right. $

Which of the following is the general solution of the homogeneous second-order differential equation are arbitrary real constants.

If Y₁ and Y₂ are both solutions of the differential equation then Y₁ - Y₂ is also a solution of this equation.

Consider this initial value problem: Which of the following is an accurate description of the long-term behavior of the solution?

Consider the nonhomogeneous differential equation What is the general solution of this differential equation?

Consider this second-order nonhomogeneous differential equation: Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C₁ and C₂ are arbitrary real constants.

Use variation of parameters to find the general solution of the nonhomogeneous differential equation

Consider this second-order nonhomogeneous differential equation: Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C₁ and C₂ are arbitrary real constants.

Which of these is the general solution of the second-order nonhomogeneous differential equation and all capital letters are arbitrary real constants.

Consider the differential equation Which of the following statements is true?

Consider this second-order nonhomogeneous differential equation: Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C₁ and C₂ are arbitrary real constants.

Use the method of reduction of order to find a second solution of the differential equation using the fact that y₁ = t^-1 is a solution. is a solution.

Which of these is the general solution of the second-order nonhomogeneous differential equation , and all capital letters are arbitrary real constants.

Use variation of parameters to find the general solution of the nonhomogeneous differential equation

Consider this second-order nonhomogeneous differential equation: Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C₁ and C₂ are arbitrary real constants.

Introduction

First-Order Differential Equations

Higher-Order Linear Differential Equations

Series Solutions of Second-Order Linear Equations

The Laplace Transform

Systems of First-Order Linear Equations

Numerical Methods

Nonlinear Differential Equations and Stability

Partial Differential Equations and Fourier Series

Boundary Value Problems and Sturm-Liouville Theory

Filters

Exam 3: Second-Order Linear Differential Equations

Which of the following is the general solution of the homogeneous second-order differential equation are arbitrary real constants.

If Y1 and Y2 are both solutions of the differential equation then Y1 - Y2 is also a solution of this equation.

For what value(s) of α\alphaα is y = a solution of the second-order homogeneous differential equation

Consider this initial value problem: Which of the following is an accurate description of the long-term behavior of the solution?

Consider the nonhomogeneous differential equation What is the general solution of this differential equation?

Consider this second-order nonhomogeneous differential equation: Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C1 and C2 are arbitrary real constants.

Use variation of parameters to find the general solution of the nonhomogeneous differential equation

Consider this second-order nonhomogeneous differential equation: Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C1 and C2 are arbitrary real constants.

Which of these is the general solution of the second-order nonhomogeneous differential equation and all capital letters are arbitrary real constants.

Consider the differential equation Which of the following statements is true?

Consider this second-order nonhomogeneous differential equation: Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C1 and C2 are arbitrary real constants.

Use the method of reduction of order to find a second solution of the differential equation using the fact that y1 = t-1 is a solution. is a solution.

Which of these is the general solution of the second-order nonhomogeneous differential equation , and all capital letters are arbitrary real constants.

Use variation of parameters to find the general solution of the nonhomogeneous differential equation

Consider this second-order nonhomogeneous differential equation: Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C1 and C2 are arbitrary real constants.

Introduction

First-Order Differential Equations

Higher-Order Linear Differential Equations

Series Solutions of Second-Order Linear Equations

The Laplace Transform

Systems of First-Order Linear Equations

Numerical Methods

Nonlinear Differential Equations and Stability

Partial Differential Equations and Fourier Series

Boundary Value Problems and Sturm-Liouville Theory

Filters

If Y₁ and Y₂ are both solutions of the differential equation then Y₁ - Y₂ is also a solution of this equation.

Consider this second-order nonhomogeneous differential equation: Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C₁ and C₂ are arbitrary real constants.

Consider this second-order nonhomogeneous differential equation: Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C₁ and C₂ are arbitrary real constants.

Consider this second-order nonhomogeneous differential equation: Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C₁ and C₂ are arbitrary real constants.

Use the method of reduction of order to find a second solution of the differential equation using the fact that y₁ = t^-1 is a solution. is a solution.

Consider this second-order nonhomogeneous differential equation: Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, C₁ and C₂ are arbitrary real constants.