Question 1

Consider the pair of functions y1 = t and y2 = 3t2.
Which of these statements are true? Select all that apply.

Accepted Answer

A)  W[y1 , y2](t) > 0 for all values of t in the interval (-2, 2). 
B)  W[y1 , y1](t) = 3t2 
C)  The pair y1 and y2 constitutes a fundamental set of solutions to some second-order differential equation of the form   on the interval (-2, 2). 
D)  Abel's theorem implies that y1 and y2 cannot both be solutions of any differential equation of the form   on the interval (-2, 2). 
E)  Since there exists a value of t0 in the interval (-2, 2) for which W[y1 ,y2 ](t) = 0, there must exist a differential equation of the form   for which the pair y1 and y2 constitute a fundamental set of solutions on the interval (-2, 2). 
A)  W[y1 , y2](t) > 0 for all values of t in the interval (-2, 2). 
B)  W[y1 , y1](t) = 3t2 
C)  The pair y1 and y2 constitutes a fundamental set of solutions to some second-order differential equation of the form   on the interval (-2, 2). 
D)  Abel's theorem implies that y1 and y2 cannot both be solutions of any differential equation of the form   on the interval (-2, 2). 
E)  Since there exists a value of t0 in the interval (-2, 2) for which W[y1 ,y2 ](t) = 0, there must exist a differential equation of the form   for which the pair y1 and y2 constitute a fundamental set of solutions on the interval (-2, 2). B, D

Question 2

Compute the Wronskian of the pair of functions sin(5t) and cos(5t).

Accepted Answer

A)  -5 
B)  -4 
C)  1 
D)  4 
E)  5 
A)  -5 
B)  -4 
C)  1 
D)  4 
E)  5 A

Question 3

Suppose a 6-lb object stretches a spring 2 feet while in equilibrium. If the object is displaced an additional 2.5 inches and is then set in motion with an initial upward velocity of -1 feet per second.
What is the natural frequency? Round your answer to the nearest hundredth radian per second.

Accepted Answer

4.00 radians per second

Question 4

For which of these differential equations is the characteristic equation given by

Accepted Answer

A)  $ y^{\prime \prime}+50=0 $ 
B)  $ y^{\prime \prime}+50 y=0 $ 
C)  $ y^{\prime \prime}-2 y^{\prime}+50=0 $ 
D)  $ y^{\prime \prime}-2 y^{\prime}+50 y=0 $ 
E)  $ \left(y^{\prime}-(1-7 i)\right)\left(y^{\prime}-(1+7 i)\right)=0 $
F) $ \left(y^{\prime}-(1-7 i) y\right)\left(y^{\prime}-(1+7 i) y\right)=0 $ 
A)  $ y^{\prime \prime}+50=0 $ 
B)  $ y^{\prime \prime}+50 y=0 $ 
C)  $ y^{\prime \prime}-2 y^{\prime}+50=0 $ 
D)  $ y^{\prime \prime}-2 y^{\prime}+50 y=0 $ 
E)  $ \left(y^{\prime}-(1-7 i)\right)\left(y^{\prime}-(1+7 i)\right)=0 $
F) $ \left(y^{\prime}-(1-7 i) y\right)\left(y^{\prime}-(1+7 i) y\right)=0 $

Question 5

Suppose that Y1 and Y2 are both solutions of the differential equation   . 
Which of the following must also be solutions of this differential equation? Select all that apply. Here, C1 , and C2 are arbitrary real constants.

Accepted Answer

A)  $ 5 y_{1}-4 y_{2} $ 
B)  $ t y_{1} $ 
C)  $ C_{1} $ 
D)  $ \left(C_{1} y_{1}\right) \cdot\left(C_{2} y_{2}\right) $ 
E)  $ C_{1}\left(y_{1}+y_{2}\right) $
F) $ C_{1}\left(7 y_{1}-9 y_{2}\right)-C_{2}\left(2 y_{1}-7 y_{2}\right. $ 
A)  $ 5 y_{1}-4 y_{2} $ 
B)  $ t y_{1} $ 
C)  $ C_{1} $ 
D)  $ \left(C_{1} y_{1}\right) \cdot\left(C_{2} y_{2}\right) $ 
E)  $ C_{1}\left(y_{1}+y_{2}\right) $
F) $ C_{1}\left(7 y_{1}-9 y_{2}\right)-C_{2}\left(2 y_{1}-7 y_{2}\right. $

Question 6

For which of these differential equations is the characteristic equation given by 6   + 7 = 0?

Accepted Answer

A)  6   + 7 = 0 
B)  6   + 7 = 0 
C)  6   + 7y = 0 
D)  6   + 7y = 0 
A)  6   + 7 = 0 
B)  6   + 7 = 0 
C)  6   + 7y = 0 
D)  6   + 7y = 0

Question 7

Consider the nonhomogeneous differential equation   Assume that   is a particular solution of this differential equation. Use variation of parameters to determine u1 (t) and u2 (t).
u1 (t) = ________ u2(t) = ________

Accepted Answer

Question 8

Which of the following are solutions to the homogeneous second-order differential equation   ? 
Select all that apply.

Accepted Answer

A)  $ y_{1}=8 e^{-2 t}+2 e^{2 t} $ 
B)  $ y_{2}=\mathrm{Ce}^{-2 t} $, where $ \mathrm{C} $ is any real constant 
C)  $ y_{3}=8\left(e^{2 t}+e^{-2 t}\right) $ 
D)  $ y_{4}=C e^{2 t} $, where $ C $ is any real constant 
E)  $ y_{5}=\left(C_{1} e^{2 t}\right) \cdot\left(C_{2} e^{-2 t}\right) $, where $ C_{1} $ and $ C_{2} $ are any real constants
F) $ y_{6}=2 e^{-2 t} $
G) $ y_{7}=C\left(e^{-2 t}+e^{2 t}\right) $, where $ C $ is any real constant 
A)  $ y_{1}=8 e^{-2 t}+2 e^{2 t} $ 
B)  $ y_{2}=\mathrm{Ce}^{-2 t} $, where $ \mathrm{C} $ is any real constant 
C)  $ y_{3}=8\left(e^{2 t}+e^{-2 t}\right) $ 
D)  $ y_{4}=C e^{2 t} $, where $ C $ is any real constant 
E)  $ y_{5}=\left(C_{1} e^{2 t}\right) \cdot\left(C_{2} e^{-2 t}\right) $, where $ C_{1} $ and $ C_{2} $ are any real constants
F) $ y_{6}=2 e^{-2 t} $
G) $ y_{7}=C\left(e^{-2 t}+e^{2 t}\right) $, where $ C $ is any real constant

Question 9

What is the general solution of the homogeneous second-order Cauchy Euler differential equation   are arbitrary real constants.

Accepted Answer

A)  $ y=C_{1} t^{-6}+C_{2} t^{6} $ 
B)  $ y=C_{1}(t \ln t)^{-6}+C_{2}(t \ln t)^{6} $ 
C)  $ y=t^{-6}\left(C_{1}+C_{2} \ln t\right) $ 
D)  $ y=C_{1} t^{-6}+C_{2}(t \ln t)^{-6} $ 
A)  $ y=C_{1} t^{-6}+C_{2} t^{6} $ 
B)  $ y=C_{1}(t \ln t)^{-6}+C_{2}(t \ln t)^{6} $ 
C)  $ y=t^{-6}\left(C_{1}+C_{2} \ln t\right) $ 
D)  $ y=C_{1} t^{-6}+C_{2}(t \ln t)^{-6} $

Question 10

Consider this second-order nonhomogeneous differential equation:
 
Which of these is a suitable form of a particular solution Y(t) of the nonhomogeneous differential equation if the method of undetermined coefficients is to be used? Here, all capital letters represent arbitrary real constants.

Accepted Answer

A)  $ Y(t)=A e^{-3 t} $ 
B)  $ Y(t)=e^{A t} $ 
C)  $ Y(t)=A e^{B t} $ 
D)  $ Y(t)=A+e^{-3 t} $ 
E)  $ Y(t)=A e^{-3 t}+B $ 
A)  $ Y(t)=A e^{-3 t} $ 
B)  $ Y(t)=e^{A t} $ 
C)  $ Y(t)=A e^{B t} $ 
D)  $ Y(t)=A+e^{-3 t} $ 
E)  $ Y(t)=A e^{-3 t}+B $

Question 11

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

Question 12

Suppose a 96-lb object stretches a spring 3 feet while in equilibrium, and a dashpot provides a damping force of 5 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 3/2 feet below the horizontal with an upward velocity of - 2 feet per second.
For what values of A, B, C, and D can the solution,   be expressed in the form . Provide the exact values, not decimal approximations.
A = ____________, B = ____________, C = ____________, tan D = ____________

Accepted Answer

The answer of Suppose a 96-lb object stretches a spring...

Question 13

For which of these differential equations is the characteristic equation given by r(10r + 1) = 0?

Accepted Answer

A)    (10   + 1) = 0 
B)  10   + 1y = 0 
C)    (10   + 1y) = 0 
D)  10   + 1   = 0 
E)  10   + 1y = 0 
A)    (10   + 1) = 0 
B)  10   + 1y = 0 
C)    (10   + 1y) = 0 
D)  10   + 1   = 0 
E)  10   + 1y = 0

Question 14

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

Question 15

Consider this initial value problem:
   
Find a particular solution of the given nonhomogeneous differential equation.

Accepted Answer

The answer of Consider this initial value problem:
  ...

Question 16

Suppose a 10-lb object stretches a spring 2.5 feet while in equilibrium. If the object is displaced an additional 0.5 inches and is then set in motion with an initial upward velocity of -0.7 feet per second.What is the amplitude, R, in feet? Round your answer to the nearest hundredth of a foot.

Accepted Answer

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

Question 17

Consider this initial value problem:
 
For what values of $\alpha$ does the solution tend to 0 as t $\rightarrow$ $\infty$?

Accepted Answer

A)  all real numbers 
B)  all nonzero real numbers 
C)  all positive real numbers 
D)  all negative real numbers 
A)  all real numbers 
B)  all nonzero real numbers 
C)  all positive real numbers 
D)  all negative real numbers

Question 18

Suppose a 160-lb object stretches a spring 12 feet while in equilibrium, and a dashpot provides a damping force of 5 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 - 3/2 feet per second.
What is the quasi-frequency? Provide the exact values, not a decimal approximation.

Accepted Answer

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

Question 19

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

Accepted Answer

A)  $ y=C_{1} e^{4 t} \sin (6 t)+C_{2} e^{4 t} \cos (6 t) $ 
B)  $ y=e^{-4 t}\left(C_{1} \sin (6 t)+C_{2} \cos (6 t)\right) $ 
C)  $ y=C_{1} e^{4 t} \cos (6 t)+C_{2} e^{4 t} \sin (6 t)+C $ 
D)  $ y=e^{6 t}(\sin (4 t)+\cos (6 t))+C $ 
E)  $ y=C_{1} e^{-4 t} \sin (6 t)+C_{2} e^{-4 t} \cos (6 t)+C $ 
A)  $ y=C_{1} e^{4 t} \sin (6 t)+C_{2} e^{4 t} \cos (6 t) $ 
B)  $ y=e^{-4 t}\left(C_{1} \sin (6 t)+C_{2} \cos (6 t)\right) $ 
C)  $ y=C_{1} e^{4 t} \cos (6 t)+C_{2} e^{4 t} \sin (6 t)+C $ 
D)  $ y=e^{6 t}(\sin (4 t)+\cos (6 t))+C $ 
E)  $ y=C_{1} e^{-4 t} \sin (6 t)+C_{2} e^{-4 t} \cos (6 t)+C $

Question 20

Suppose a 128-lb object stretches a spring 3 feet while in equilibrium, and a dashpot provides a damping force of 5 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 feet per second.
For what values of the arbitrary constants C1 and C2 does a general solution of the form y(t) =   satisfy the initial conditions? Provide the exact values, not decimal approximations.
C1__________, C2____________

Accepted Answer

Consider the pair of functions y₁ = t and y₂ = 3t². Which of these statements are true? Select all that apply.

Compute the Wronskian of the pair of functions sin(5t) and cos(5t).

Suppose a 6-lb object stretches a spring 2 feet while in equilibrium. If the object is displaced an additional 2.5 inches and is then set in motion with an initial upward velocity of -1 feet per second. What is the natural frequency? Round your answer to the nearest hundredth radian per second.

For which of these differential equations is the characteristic equation given by

Suppose that Y₁ and Y₂ are both solutions of the differential equation . Which of the following must also be solutions of this differential equation? Select all that apply. Here, C₁ , and C₂ are arbitrary real constants.

For which of these differential equations is the characteristic equation given by 6 + 7 = 0?

Consider the nonhomogeneous differential equation Assume that is a particular solution of this differential equation. Use variation of parameters to determine u₁ (t) and u₂ (t). u₁ (t) = u₂(t) =

Which of the following are solutions to the homogeneous second-order differential equation ? Select all that apply.

What is the general solution of the homogeneous second-order Cauchy Euler differential equation are arbitrary real constants.

Consider this second-order nonhomogeneous differential equation: Which of these is a suitable form of a particular solution Y(t) of the nonhomogeneous differential equation if the method of undetermined coefficients is to be used? Here, all capital letters represent 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.

For which of these differential equations is the characteristic equation given by r(10r + 1) = 0?

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

Consider this initial value problem: Find a particular solution of the given nonhomogeneous differential equation.

Suppose a 10-lb object stretches a spring 2.5 feet while in equilibrium. If the object is displaced an additional 0.5 inches and is then set in motion with an initial upward velocity of -0.7 feet per second.What is the amplitude, R, in feet? Round your answer to the nearest hundredth of a foot.

Consider this initial value problem: $Consider this initial value problem: For what values of \alpha does the solution tend to 0 as t \rightarrow \infty ?$ For what values of $\alpha$ does the solution tend to 0 as t $\rightarrow$ $\infty$ ?

Which of the following is the general solution of the homogeneous second-order differential equation 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

Consider the pair of functions y1 = t and y2 = 3t2. Which of these statements are true? Select all that apply.

Compute the Wronskian of the pair of functions sin(5t) and cos(5t).

Suppose a 6-lb object stretches a spring 2 feet while in equilibrium. If the object is displaced an additional 2.5 inches and is then set in motion with an initial upward velocity of -1 feet per second. What is the natural frequency? Round your answer to the nearest hundredth radian per second.

For which of these differential equations is the characteristic equation given by

Suppose that Y1 and Y2 are both solutions of the differential equation . Which of the following must also be solutions of this differential equation? Select all that apply. Here, C1 , and C2 are arbitrary real constants.

For which of these differential equations is the characteristic equation given by 6 + 7 = 0?

Consider the nonhomogeneous differential equation Assume that is a particular solution of this differential equation. Use variation of parameters to determine u1 (t) and u2 (t). u1 (t) = ________ u2(t) = ________

Which of the following are solutions to the homogeneous second-order differential equation ? Select all that apply.

What is the general solution of the homogeneous second-order Cauchy Euler differential equation are arbitrary real constants.

Consider this second-order nonhomogeneous differential equation: Which of these is a suitable form of a particular solution Y(t) of the nonhomogeneous differential equation if the method of undetermined coefficients is to be used? Here, all capital letters represent 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.

For which of these differential equations is the characteristic equation given by r(10r + 1) = 0?

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

Consider this initial value problem: Find a particular solution of the given nonhomogeneous differential equation.

Suppose a 10-lb object stretches a spring 2.5 feet while in equilibrium. If the object is displaced an additional 0.5 inches and is then set in motion with an initial upward velocity of -0.7 feet per second.What is the amplitude, R, in feet? Round your answer to the nearest hundredth of a foot.

Consider this initial value problem: For what values of α\alphaα does the solution tend to 0 as t →\rightarrow→ ∞\infty∞ ?

Which of the following is the general solution of the homogeneous second-order differential equation 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

Consider the pair of functions y₁ = t and y₂ = 3t². Which of these statements are true? Select all that apply.

Suppose that Y₁ and Y₂ are both solutions of the differential equation . Which of the following must also be solutions of this differential equation? Select all that apply. Here, C₁ , and C₂ are arbitrary real constants.

Consider the nonhomogeneous differential equation Assume that is a particular solution of this differential equation. Use variation of parameters to determine u₁ (t) and u₂ (t). u₁ (t) = u₂(t) =

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 initial value problem: $Consider this initial value problem: For what values of \alpha does the solution tend to 0 as t \rightarrow \infty ?$ For what values of $\alpha$ does the solution tend to 0 as t $\rightarrow$ $\infty$ ?