Question 1

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

Accepted Answer

A)  $ y=C_{1} e^{-\frac{3}{4} t}+C_{2} t e^{-\frac{3}{4} t} $ 
B)  $ y=C_{1} e^{-\frac{3}{4} t}+C_{2} e^{\frac{3}{4} t} $ 
C)  $ y=C_{1} e^{\frac{3}{4} t}+C_{2} t e^{\frac{3}{4} t} $ 
D)  $ y=C_{1} e^{-\frac{4}{3} t}+C_{2} t e^{-\frac{4}{3} t} $ 
E)  $ y=C_{1} t e^{-\frac{3}{4} t}+C_{2} $ 
A)  $ y=C_{1} e^{-\frac{3}{4} t}+C_{2} t e^{-\frac{3}{4} t} $ 
B)  $ y=C_{1} e^{-\frac{3}{4} t}+C_{2} e^{\frac{3}{4} t} $ 
C)  $ y=C_{1} e^{\frac{3}{4} t}+C_{2} t e^{\frac{3}{4} t} $ 
D)  $ y=C_{1} e^{-\frac{4}{3} t}+C_{2} t e^{-\frac{4}{3} t} $ 
E)  $ y=C_{1} t e^{-\frac{3}{4} t}+C_{2} $

Question 2

Compute the Wronskian of the pair of functions 2t   and 4   .

Accepted Answer

The answer of Compute the Wronskian of the pair of...

Question 3

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

Accepted Answer

A)  $ \left.y(t)=C_{1} \cos (5 t)+C_{2} \sin (5 t)+\frac{1}{25} \cos ^{2}(5 t)-\frac{1}{75} \cos ^{4} 5 t\right)+\frac{1}{75} \sin ^{4}(5 t) $ 
B)  $ y(t)=C_{1} \cos (5 t)+C_{2} \sin (5 t)+\cos ^{2}(5 t)-\frac{1}{3} \cos ^{4}(5 t)+\frac{1}{3} \sin ^{4}(5 t) $ 
C)  $ y(t)=C_{1} \cos (5 t)+C_{2} \sin (5 t)+\frac{1}{5} \sin (5 t) \cos (5 t)\left[1-\frac{1}{3} \cos ^{2}(5 t)\right]+\frac{1}{15} \sin ^{3}(5 t) \cos (5 t) $ 
D)  $ y(t)=C_{1} \cos (5 t)+C_{2} \sin (5 t)+\sin (5 t) \cos (5 t)\left[1-\frac{1}{3} \cos ^{2}(5 t)\right)+\frac{1}{3} \sin ^{3}(5 t) \cos (5 t) $ 
A)  $ \left.y(t)=C_{1} \cos (5 t)+C_{2} \sin (5 t)+\frac{1}{25} \cos ^{2}(5 t)-\frac{1}{75} \cos ^{4} 5 t\right)+\frac{1}{75} \sin ^{4}(5 t) $ 
B)  $ y(t)=C_{1} \cos (5 t)+C_{2} \sin (5 t)+\cos ^{2}(5 t)-\frac{1}{3} \cos ^{4}(5 t)+\frac{1}{3} \sin ^{4}(5 t) $ 
C)  $ y(t)=C_{1} \cos (5 t)+C_{2} \sin (5 t)+\frac{1}{5} \sin (5 t) \cos (5 t)\left[1-\frac{1}{3} \cos ^{2}(5 t)\right]+\frac{1}{15} \sin ^{3}(5 t) \cos (5 t) $ 
D)  $ y(t)=C_{1} \cos (5 t)+C_{2} \sin (5 t)+\sin (5 t) \cos (5 t)\left[1-\frac{1}{3} \cos ^{2}(5 t)\right)+\frac{1}{3} \sin ^{3}(5 t) \cos (5 t) $

Question 4

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

Accepted Answer

A)    + 4   - 21y = 0 
B)  (   - 3)(   + 7) = 0 
C)    + 4   - 21 = 0 
D)    - 4   - 21 = 0 
A)    + 4   - 21y = 0 
B)  (   - 3)(   + 7) = 0 
C)    + 4   - 21 = 0 
D)    - 4   - 21 = 0

Question 5

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

Accepted Answer

Question 6

Consider the homogeneous second-order Cauchy Euler differential equation
   
For what value α does the solution of the initial value problem comprised of this differential equation and the initial conditions y(1) = α,   (1) = 6 tend to 0 as t → ∞? Enter your answer as a simplified fraction. If there is no such value of α, enter 'none'.

Accepted Answer

The answer of Consider the homogeneous second-order Cauchy Euler differential...

Question 7

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)=\left(e^{-6 t}+e^{-2 t}\right) \cdot(A t+
B)  $ 
B)  $ Y(t)=A t+B $ 
C)  $ Y(t)=A t $ 
D)  $ Y(t)=A t+e^{-6 t}+e^{-2 t} $ 
A)  $ Y(t)=\left(e^{-6 t}+e^{-2 t}\right) \cdot(A t+
B)  $ 
B)  $ Y(t)=A t+B $ 
C)  $ Y(t)=A t $ 
D)  $ Y(t)=A t+e^{-6 t}+e^{-2 t} $

Question 8

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(e^{5 t}+e^{10 t}\right) $ 
B)  $ y=C_{1} e^{-5 t}+C_{2} e^{-10 t} $ 
C)  $ y=C_{1} e^{5 t}+C_{2} e^{10 t} $ 
D)  $ y=C\left(e^{-5 t}+e^{-10 t}\right) $ 
E)  $ y=C_{1} e^{-5 t}+C_{2} e^{-10 t}+y+\left(C_{1} e^{-5 t}\right) \cdot\left(C_{2} e^{-10 t}\right) $
F) $ y=\left(C_{1} e^{-5 t}\right) \cdot\left(C_{2} e^{-10 t}\right) $ 
A)  $ y=C\left(e^{5 t}+e^{10 t}\right) $ 
B)  $ y=C_{1} e^{-5 t}+C_{2} e^{-10 t} $ 
C)  $ y=C_{1} e^{5 t}+C_{2} e^{10 t} $ 
D)  $ y=C\left(e^{-5 t}+e^{-10 t}\right) $ 
E)  $ y=C_{1} e^{-5 t}+C_{2} e^{-10 t}+y+\left(C_{1} e^{-5 t}\right) \cdot\left(C_{2} e^{-10 t}\right) $
F) $ y=\left(C_{1} e^{-5 t}\right) \cdot\left(C_{2} e^{-10 t}\right) $

Question 9

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

Accepted Answer

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

Question 10

Consider the initial value problem
   
What is the t-coordinate of the local extreme value of y = y(t) on the interval (0, ∞)? Enter your answer as a decimal accurate to three decimal places.

Accepted Answer

The answer of Consider the initial value problem
  ...

Question 11

Compute the Wronskian of the pair of functions

Accepted Answer

A)  -2   
B)  -6   
C)  -8   
D)  -6 
E)  -8   
A)  -2   
B)  -6   
C)  -8   
D)  -6 
E)  -8

Question 12

What is the solution of this initial value problem:

Accepted Answer

A)  $ y=3 \cos (11 t)+\frac{10}{11} \sin (11 t) $ 
B)  $ y=\cos (11 t)+\sin (11 t) $ 
C)  $ y=3 \sin (11 t)+10 \cos (11 t) $ 
D)  $ y=10 e^{-11 t}+3 e^{-11 t} $ 
E)  $ y=3 e^{121 t}+\frac{10}{11} e^{-121 t} $ 
A)  $ y=3 \cos (11 t)+\frac{10}{11} \sin (11 t) $ 
B)  $ y=\cos (11 t)+\sin (11 t) $ 
C)  $ y=3 \sin (11 t)+10 \cos (11 t) $ 
D)  $ y=10 e^{-11 t}+3 e^{-11 t} $ 
E)  $ y=3 e^{121 t}+\frac{10}{11} e^{-121 t} $

Question 13

Consider this second-order nonhomogeneous differential equation:
   
What 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? Use capital letters to represent arbitrary real constants.

Accepted Answer

The answer of Consider this second-order nonhomogeneous differential equation:
 ...

Question 14

Consider the initial value problem
 
What is the solution of this initial value problem?

Accepted Answer

A)  $ y=\left(\frac{\alpha}{2}-\frac{8}{7}\right) e^{\frac{7}{4} t}+\left(\frac{\alpha}{2}+\frac{8}{7}\right) e^{\frac{7}{4} t} $ 
B)  $ y=\left(\frac{\alpha}{2}+\frac{8}{7}\right) e^{\frac{7}{4} t}+\left(\frac{\alpha}{2}-\frac{8}{7}\right) e^{\frac{7}{4} t} $ 
C)  $ y=\left(\frac{\alpha}{2}+\frac{8}{7}\right) e^{\frac{4}{7} t}+\left(\frac{\alpha}{2}-\frac{8}{7}\right) e^{\frac{4}{7} t} $ 
D)  $ y=\left(\frac{a}{2}-\frac{8}{7}\right) e^{\frac{4}{7} t}+\left(\frac{a}{2}+\frac{8}{7}\right) e^{\frac{4}{7} t} $ 
A)  $ y=\left(\frac{\alpha}{2}-\frac{8}{7}\right) e^{\frac{7}{4} t}+\left(\frac{\alpha}{2}+\frac{8}{7}\right) e^{\frac{7}{4} t} $ 
B)  $ y=\left(\frac{\alpha}{2}+\frac{8}{7}\right) e^{\frac{7}{4} t}+\left(\frac{\alpha}{2}-\frac{8}{7}\right) e^{\frac{7}{4} t} $ 
C)  $ y=\left(\frac{\alpha}{2}+\frac{8}{7}\right) e^{\frac{4}{7} t}+\left(\frac{\alpha}{2}-\frac{8}{7}\right) e^{\frac{4}{7} t} $ 
D)  $ y=\left(\frac{a}{2}-\frac{8}{7}\right) e^{\frac{4}{7} t}+\left(\frac{a}{2}+\frac{8}{7}\right) e^{\frac{4}{7} t} $

Question 15

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

Accepted Answer

A)  $ y(t)=C_{1} e^{-2 t}+C_{1} e^{-3 t}-\frac{1}{t} $ 
B)  $ y(t)=C_{1} e^{2 t}+C_{1} e^{3 t}-\frac{1}{t} $ 
C)  $ y(t)=C_{1} t^{2}+C_{1} t^{3}+\frac{6}{5 t^{2}} $ 
D)  $ y(t)=C_{1} t^{2}+C_{1} t^{3}+\frac{1}{20 t^{2}} $ 
E)  $ y(t)=C_{1} t^{2}+C_{1} t^{3}+\frac{1}{20 t^{2}} $. 
A)  $ y(t)=C_{1} e^{-2 t}+C_{1} e^{-3 t}-\frac{1}{t} $ 
B)  $ y(t)=C_{1} e^{2 t}+C_{1} e^{3 t}-\frac{1}{t} $ 
C)  $ y(t)=C_{1} t^{2}+C_{1} t^{3}+\frac{6}{5 t^{2}} $ 
D)  $ y(t)=C_{1} t^{2}+C_{1} t^{3}+\frac{1}{20 t^{2}} $ 
E)  $ y(t)=C_{1} t^{2}+C_{1} t^{3}+\frac{1}{20 t^{2}} $.

Question 16

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

Accepted Answer

A)  $ y_{1}=C e^{-\frac{4}{3} t^{2}} $, where $ C $ is any real constant 
B)  $ y_{2}=-4 e^{-\frac{4}{3} t}+3 e^{\frac{4}{3} t} $ 
C)  $ y_{3}=C e^{\frac{3}{4} t} $, where $ C $ is any real constant 
D)  $ y_{4}=C\left(e^{-\frac{4}{3} t}+e^{\frac{4}{3} t}\right) $, where $ C $ is any real constant 
E)  $ y_{1}=3 e^{\frac{3}{4} t}+-4 e^{-\frac{3}{4} t} $
F) $ y_{6}=t e^{3} $ 
A)  $ y_{1}=C e^{-\frac{4}{3} t^{2}} $, where $ C $ is any real constant 
B)  $ y_{2}=-4 e^{-\frac{4}{3} t}+3 e^{\frac{4}{3} t} $ 
C)  $ y_{3}=C e^{\frac{3}{4} t} $, where $ C $ is any real constant 
D)  $ y_{4}=C\left(e^{-\frac{4}{3} t}+e^{\frac{4}{3} t}\right) $, where $ C $ is any real constant 
E)  $ y_{1}=3 e^{\frac{3}{4} t}+-4 e^{-\frac{3}{4} t} $
F) $ y_{6}=t e^{3} $

Question 17

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 used? Here, all capital letters represent arbitrary real constants.

Accepted Answer

A)  $ Y(t)=A t+B t^{4} $ 
B)  $ Y(t)=\left(A t+B t^{4}\right) e^{4 t} \sin (2 t)+\left(C t+D t^{4}\right) e^{4 t} \cos (2 t) $ 
C)  $ Y(t)=A t^{4}+B t^{3}+C t^{2}+D t+E $ 
D)  $ Y(t)=\left(A t^{4}+B t^{3}+C t^{2}+D t+E\right) e^{4 t}(\sin (2 t)+\cos (2 t)) $ 
E)  $ Y(t)=\left(A t^{4}+B t\right) e^{2 t} \sin (4 t)+\left(C t^{4}+D t\right) e^{2 t} \cos (4 t) $
F) $ Y(t)=\left(A t^{4}+B t^{3}+C t^{2}+D t+E\right) e^{2 t}(\sin (4 t)+\cos (4 t)) $ 
A)  $ Y(t)=A t+B t^{4} $ 
B)  $ Y(t)=\left(A t+B t^{4}\right) e^{4 t} \sin (2 t)+\left(C t+D t^{4}\right) e^{4 t} \cos (2 t) $ 
C)  $ Y(t)=A t^{4}+B t^{3}+C t^{2}+D t+E $ 
D)  $ Y(t)=\left(A t^{4}+B t^{3}+C t^{2}+D t+E\right) e^{4 t}(\sin (2 t)+\cos (2 t)) $ 
E)  $ Y(t)=\left(A t^{4}+B t\right) e^{2 t} \sin (4 t)+\left(C t^{4}+D t\right) e^{2 t} \cos (4 t) $
F) $ Y(t)=\left(A t^{4}+B t^{3}+C t^{2}+D t+E\right) e^{2 t}(\sin (4 t)+\cos (4 t)) $

Question 18

Consider the initial value problem
 
On which of these intervals is this initial value problem certain to have a unique twice differentiable solution? Select all that apply.

Accepted Answer

A)  $ (-4,-3.5) $ 
B)  $ \left(\frac{11}{32}, \frac{25}{32}\right) $ 
C)  $ (0,1) $ 
D)  $ \left(\frac{1}{8}, 1\right) $ 
E)  $ (-\infty, \infty) $ 
A)  $ (-4,-3.5) $ 
B)  $ \left(\frac{11}{32}, \frac{25}{32}\right) $ 
C)  $ (0,1) $ 
D)  $ \left(\frac{1}{8}, 1\right) $ 
E)  $ (-\infty, \infty) $

Question 19

What is the solution of the initial value problem

Accepted Answer

A)  $ y=-\frac{4}{3}-\frac{2}{3} $ 
B)  $ y=-2 t+e^{-3 t} $ 
C)  $ y=\frac{2}{3} e^{-3 t}-\frac{11}{3} $ 
D)  $ y=-\frac{2}{3} e^{-3 t}+\frac{11}{3} $ 
A)  $ y=-\frac{4}{3}-\frac{2}{3} $ 
B)  $ y=-2 t+e^{-3 t} $ 
C)  $ y=\frac{2}{3} e^{-3 t}-\frac{11}{3} $ 
D)  $ y=-\frac{2}{3} e^{-3 t}+\frac{11}{3} $

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}\left(e^{10 t}+e^{6 t}\right)+C_{2} $ 
B)  $ y(t)=C_{1} e^{-10 t}+C_{2} e^{6 t} $ 
C)  $ y(t)=C_{1} e^{-10 t}+C_{2} e^{-6 t} $ 
D)  $ y(t)=C_{1} e^{10 t}+C_{2} e^{6 t} $ 
E)  $ y(t)=C_{1} e^{-10 t}+C_{2}\left(e^{-6 t}+t\right) $ 
A)  $ y(t)=C_{1}\left(e^{10 t}+e^{6 t}\right)+C_{2} $ 
B)  $ y(t)=C_{1} e^{-10 t}+C_{2} e^{6 t} $ 
C)  $ y(t)=C_{1} e^{-10 t}+C_{2} e^{-6 t} $ 
D)  $ y(t)=C_{1} e^{10 t}+C_{2} e^{6 t} $ 
E)  $ y(t)=C_{1} e^{-10 t}+C_{2}\left(e^{-6 t}+t\right) $

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

Compute the Wronskian of the pair of functions 2t and 4 .

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

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

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

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.

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

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

Consider the initial value problem What is the t-coordinate of the local extreme value of y = y(t) on the interval (0, ∞)? Enter your answer as a decimal accurate to three decimal places.

Compute the Wronskian of the pair of functions

What is the solution of this initial value problem:

Consider this second-order nonhomogeneous differential equation: What 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? Use capital letters to represent arbitrary real constants.

Consider the initial value problem What is the solution of this initial value problem?

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

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

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 used? Here, all capital letters represent arbitrary real constants.

Consider the initial value problem On which of these intervals is this initial value problem certain to have a unique twice differentiable solution? Select all that apply.

What is the solution of the initial value problem

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 arbitrary real constants.

Compute the Wronskian of the pair of functions 2t and 4 .

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

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

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

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.

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

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

Consider the initial value problem What is the t-coordinate of the local extreme value of y = y(t) on the interval (0, ∞)? Enter your answer as a decimal accurate to three decimal places.

Compute the Wronskian of the pair of functions

What is the solution of this initial value problem:

Consider this second-order nonhomogeneous differential equation: What 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? Use capital letters to represent arbitrary real constants.

Consider the initial value problem What is the solution of this initial value problem?

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

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

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 used? Here, all capital letters represent arbitrary real constants.

Consider the initial value problem On which of these intervals is this initial value problem certain to have a unique twice differentiable solution? Select all that apply.

What is the solution of the initial value problem

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

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