Question 1

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

Accepted Answer

A)  $ y=2 \sin \left(\frac{t}{6}\right)+12 \cos \left(\frac{t}{6}\right) $ 
B)  $ y=-12 \cos \left(\frac{t}{6}\right)-2 \sin \left(\frac{t}{6}\right) $ 
C)  $ y=2 \cos (6 t)-2 \sin (6 t) $ 
D)  $ y=-2 \cos (6 t)-2 \sin (6 t) $ 
A)  $ y=2 \sin \left(\frac{t}{6}\right)+12 \cos \left(\frac{t}{6}\right) $ 
B)  $ y=-12 \cos \left(\frac{t}{6}\right)-2 \sin \left(\frac{t}{6}\right) $ 
C)  $ y=2 \cos (6 t)-2 \sin (6 t) $ 
D)  $ y=-2 \cos (6 t)-2 \sin (6 t) $

Question 2

Consider the initial value problem
 
Which of these statements are true? Select all that apply.

Accepted Answer

A)  There exists a nonzero real number r such that y(t) =   is a solution of the initial value problem. 
B)  This initial value problem has only one solution on the interval (-7, 5). 
C)  The constant function y(t) = -1 is a solution of this initial value problem for all real numbers t. 
D)  There must exist a function y = ?(t) that satisfies this initial value problem on the interval   . 
E)  The constant function y(t) = 0 is the unique solution of this initial value problem on the interval   . 
A)  There exists a nonzero real number r such that y(t) =   is a solution of the initial value problem. 
B)  This initial value problem has only one solution on the interval (-7, 5). 
C)  The constant function y(t) = -1 is a solution of this initial value problem for all real numbers t. 
D)  There must exist a function y = ?(t) that satisfies this initial value problem on the interval   . 
E)  The constant function y(t) = 0 is the unique solution of this initial value problem on the interval   .

Question 3

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

Accepted Answer

A)  $ y=t^{12}+t $ 
B)  $ y=\mathrm{Cr}^{-12} $, where $ \mathrm{C} $ is any real constant 
C)  $ y=16 t $ 
D)  $ y=C\left(\frac{1}{t^{12}}+t\right) $, where $ C $ is any real constant 
E)  $ y=-9 r^{12}+C $, where $ C $ is any real constant
F) $ y=C_{1} t^{12}+C_{2} t+C $, where $ C_{3} C_{1} $, and $ C_{2} $ are arbitrary real constants 
A)  $ y=t^{12}+t $ 
B)  $ y=\mathrm{Cr}^{-12} $, where $ \mathrm{C} $ is any real constant 
C)  $ y=16 t $ 
D)  $ y=C\left(\frac{1}{t^{12}}+t\right) $, where $ C $ is any real constant 
E)  $ y=-9 r^{12}+C $, where $ C $ is any real constant
F) $ y=C_{1} t^{12}+C_{2} t+C $, where $ C_{3} C_{1} $, and $ C_{2} $ are arbitrary real constants

Question 4

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) = 4 remain bounded as   ? 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 5

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

Question 6

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 \sin (5 t)+B e^{5 t}+C $ 
B)  $ Y(t)=(A t+
B)  \sin (5 t)+C e^{5 t}+D $ 
C)  $ Y(t)=e^{5 t}(A \sin (5 t)+B \cos (5 t)+
C) +D $ 
D)  $ Y(t)=A \sin (5 t)+B \cos (5 t)+C e^{5 t}+D $ 
E)  $ Y(t)=(A t+
B)  \sin (5 t)+(C t+
D)  \cos (5 t)+E e^{5 t}+F $ 
A)  $ Y(t)=A \sin (5 t)+B e^{5 t}+C $ 
B)  $ Y(t)=(A t+
B)  \sin (5 t)+C e^{5 t}+D $ 
C)  $ Y(t)=e^{5 t}(A \sin (5 t)+B \cos (5 t)+
C) +D $ 
D)  $ Y(t)=A \sin (5 t)+B \cos (5 t)+C e^{5 t}+D $ 
E)  $ Y(t)=(A t+
B)  \sin (5 t)+(C t+
D)  \cos (5 t)+E e^{5 t}+F $

Question 7

Consider the homogeneous second-order Cauchy Euler differential equation
 
What is the general solution of this differential equation? Here, C1 and C2 are arbitrary real constants.

Accepted Answer

A)  $ y=C_{1} t^{10}+C_{2} t^{6} $ 
B)  $ y=C_{1} t^{10}+C_{2} t^{6} $. 
C)  $ y=C_{1} r^{10 t}+C_{2} t^{6 t} $ 
D)  $ y=C_{1} t^{10 t}+C_{2} f^{6 t} $. 
E)  $ y=C_{1} t^{-5}+C_{2} t^{-6} $ 
A)  $ y=C_{1} t^{10}+C_{2} t^{6} $ 
B)  $ y=C_{1} t^{10}+C_{2} t^{6} $. 
C)  $ y=C_{1} r^{10 t}+C_{2} t^{6 t} $ 
D)  $ y=C_{1} t^{10 t}+C_{2} f^{6 t} $. 
E)  $ y=C_{1} t^{-5}+C_{2} t^{-6} $

Question 8

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 9

A 6-kilogram object stretches a spring 6 centimeters while in equilibrium. The object is acted upon by an external force of 10 cos   Newtons, and moves in a medium that imparts a viscous force of 1.6 Newtons when the speed of the mass is 3.6 centimeters per second. The object is set into motion from its equilibrium position with an initial velocity of 2.9 centimeters per second. Identify the parameters A, B, C, D, E, and F so that this spring-mass system is described by the following initial value problem:
   
Round all numerical values to the nearest hundredth, if needed.

Accepted Answer

The answer of A 6-kilogram object stretches a spring 6...

Question 10

For which of the following values of r is y = C   a solution of the second-order homogeneous differential equation 4   + y = 0? Select all that apply.

Accepted Answer

A)  $ -\frac{1}{4} $ 
B)  -4 
C)  0 
D)  $ \frac{1}{4} $ 
E)  4 
A)  $ -\frac{1}{4} $ 
B)  -4 
C)  0 
D)  $ \frac{1}{4} $ 
E)  4

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

Question 12

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

Accepted Answer

A)  y(t) decreases to 0 as t$\rightarrow$ $\infty$. 
B)  y(t) is periodic with period 20$\pi$. 
C)  y(t) oscillates toward 0 as t $\rightarrow$ $\infty$. 
D)  y(t) becomes unbounded in both the positive and negative y-directions as t $\rightarrow$ $\infty$. 
A)  y(t) decreases to 0 as t$\rightarrow$ $\infty$. 
B)  y(t) is periodic with period 20$\pi$. 
C)  y(t) oscillates toward 0 as t $\rightarrow$ $\infty$. 
D)  y(t) becomes unbounded in both the positive and negative y-directions as t $\rightarrow$ $\infty$.

Question 13

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

Accepted Answer

A)  $ y=4+\mathrm{Ce}^{-6 t} $ 
B)  $ y=C_{1} e^{-4 t}+C_{2} e^{-6 t} $ 
C)  $ y=C_{1}+C_{2} e^{-6 t} $ 
D)  $ y=C_{1}+C_{2} e^{6 t} $ 
E)  $ y=4+C e^{6 t} $ 
A)  $ y=4+\mathrm{Ce}^{-6 t} $ 
B)  $ y=C_{1} e^{-4 t}+C_{2} e^{-6 t} $ 
C)  $ y=C_{1}+C_{2} e^{-6 t} $ 
D)  $ y=C_{1}+C_{2} e^{6 t} $ 
E)  $ y=4+C e^{6 t} $

Question 14

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

Question 15

What is the characteristic equation for the second-order homogeneous differential equation   ?

Accepted Answer

A)  9   - 14r = 0 
B)  9   + 12r - 26 = 0 
C)  9   - 14   = 0 
D)  9   + 12   - 26r = 0 
A)  9   - 14r = 0 
B)  9   + 12r - 26 = 0 
C)  9   - 14   = 0 
D)  9   + 12   - 26r = 0

Question 16

Consider the homogeneous second-order Cauchy Euler differential equation
   
What is the solution of the initial value problem comprised of this differential equation and the initial conditions y(1) = α,   (1) = 4?

Accepted Answer

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

Question 17

Consider the pair of functions y1 = ln t and y1 = t ln t.
Which of these statements is true?

Accepted Answer

A)  Both y1 and y2 can be solutions of the differential equation   on the interval (0, $\infty$), where p(t) and q(t) are continuous on (0, $\infty$). 
B)  The Wronskian for this function pair is strictly positive on (0, $\infty$). 
C)  Abel's theorem implies that y1 and y2 cannot both be solutions of any differential equation of the form   on the interval (0, $\infty$). 
D)  The pair y1 and y2 constitutes a fundamental set of solutions to some second-order differential equation of the form   on the interval (0, $\infty$). 
A)  Both y1 and y2 can be solutions of the differential equation   on the interval (0, $\infty$), where p(t) and q(t) are continuous on (0, $\infty$). 
B)  The Wronskian for this function pair is strictly positive on (0, $\infty$). 
C)  Abel's theorem implies that y1 and y2 cannot both be solutions of any differential equation of the form   on the interval (0, $\infty$). 
D)  The pair y1 and y2 constitutes a fundamental set of solutions to some second-order differential equation of the form   on the interval (0, $\infty$).

Question 18

The pair of functions   
forms a fundamental set of solutions for the differential equation

Accepted Answer

A) True 
 B)False

Question 19

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

Accepted Answer

A)  $ y_{1}=e^{-\frac{3}{2} t}+e^{\frac{3}{2} t} $ 
B)  $ y_{2}=-6 t e^{\frac{3}{2} t}+8 $ 
C)  $ y_{3}=C_{1} e^{-\frac{3}{2} t}+C_{2} t e^{-\frac{3}{2} t} $, where $ C_{1} $ and $ C_{2} $ are arbitrary real constants 
D)  $ y_{4}=8 e^{\frac{3}{2} t} $ 
E)  $ y_{5}=C e^{\frac{3}{2} t}+10 t e^{\frac{3}{2} t} $
F) $ y_{6}=2 e^{\frac{3}{2} t}+8 t e^{\frac{3}{2} t}+8 $42_00 
A)  $ y_{1}=e^{-\frac{3}{2} t}+e^{\frac{3}{2} t} $ 
B)  $ y_{2}=-6 t e^{\frac{3}{2} t}+8 $ 
C)  $ y_{3}=C_{1} e^{-\frac{3}{2} t}+C_{2} t e^{-\frac{3}{2} t} $, where $ C_{1} $ and $ C_{2} $ are arbitrary real constants 
D)  $ y_{4}=8 e^{\frac{3}{2} t} $ 
E)  $ y_{5}=C e^{\frac{3}{2} t}+10 t e^{\frac{3}{2} t} $
F) $ y_{6}=2 e^{\frac{3}{2} t}+8 t e^{\frac{3}{2} t}+8 $42_00

Question 20

Suppose a 6-lb object stretches a spring 1.75 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.
For what values of α, β, and γ is this spring-mass system expressed as the following initial value problem, where u(t) is the position of the object at any time t? Round your answer to three decimal places.

Accepted Answer

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

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

Consider the initial value problem Which of these statements are true? Select all that apply.

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

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

Consider the homogeneous second-order Cauchy Euler differential equation What is the general solution of this differential equation? Here, C₁ and C₂ are arbitrary real constants.

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.

For which of the following values of r is y = C a solution of the second-order homogeneous differential equation 4 + y = 0? Select all that apply.

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

Which of the following is the general solution of the homogeneous second-order 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 used? Here, all capital letters represent arbitrary real constants.

What is the characteristic equation for the second-order homogeneous differential equation ?

Consider the homogeneous second-order Cauchy Euler differential equation What is the solution of the initial value problem comprised of this differential equation and the initial conditions y(1) = α, (1) = 4?

Consider the pair of functions y₁ = ln t and y₁ = t ln t. Which of these statements is true?

The pair of functions forms a fundamental set of solutions for the differential equation

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

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

Consider the initial value problem Which of these statements are true? Select all that apply.

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

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

Consider the homogeneous second-order Cauchy Euler differential equation What is the general solution of this differential equation? Here, C1 and C2 are arbitrary real constants.

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.

For which of the following values of r is y = C a solution of the second-order homogeneous differential equation 4 + y = 0? Select all that apply.

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

Which of the following is the general solution of the homogeneous second-order 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 used? Here, all capital letters represent arbitrary real constants.

What is the characteristic equation for the second-order homogeneous differential equation ?

Consider the homogeneous second-order Cauchy Euler differential equation What is the solution of the initial value problem comprised of this differential equation and the initial conditions y(1) = α, (1) = 4?

Consider the pair of functions y1 = ln t and y1 = t ln t. Which of these statements is true?

The pair of functions forms a fundamental set of solutions for the differential equation

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

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 homogeneous second-order Cauchy Euler differential equation What is the general solution of this 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 the pair of functions y₁ = ln t and y₁ = t ln t. Which of these statements is true?