Deck 3: Second-Order Linear Differential Equations

Question 1

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

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

Answer 1

9 + 12r - 26 = 0

Question 2

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

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

Answer 2

10 + 1 = 0

Question 3

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

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

Answer 3

6 + 7y = 0

Question 4

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

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

Question 5

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

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 6

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

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 7

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

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 8

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

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 9

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.

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

Question 10

What is the solution of the initial value problem

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 11

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

A) $y=C_{1} e^{-4 t}+C_{2} e^{-8 t}$
B) $y=C_{1} e^{4 t}+C_{2} e^{8 t}$
C) $y=C_{1} e^{t}+C_{2} e^{32 t}$
D) $y=C_{1} e^{-t}+C_{2} e^{-32 t}$
E) $y=C_{1} e^{4 t}+C_{2} e^{-32 t}$

Question 12

Consider the second-order homogeneous differential equation
If the differential equation is equipped with the initial conditions
what is the solution of the resulting initial value problem?

A) $y=9 e^{2 t}-3 e^{4 t}$
B) $y=6 e^{2 t}+6 e^{4 t}$
C) $y=-9 e^{-2 t}-3 e^{-4 t}$
D) $y=6 e^{-2 t}+6 e^{-4 t}$

Question 13

Consider the initial value problem

What is the solution of this initial value problem?

A) y = -4 + 2
B) y = -4 + 2
C) y = -2 + 2
D) y = -2 - 2
E) y = -4t - 2 F) y = -2t - 2

Question 14

Consider the initial value problem

Fill in the blank: ________

Question 15

Consider the initial value problem

What is the solution of this initial value problem?

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 16

Consider the initial value problem

For what value of ? does the solution of this initial value problem tend to zero as t $\rightarrow$ $\infty$ ?

A) - 6
B) -
C) 0
D) 6
E)

Question 17

Consider the initial value problem

What is the solution of this initial value problem?

A) $y=\frac{20}{3} e^{-\frac{3}{2} t}-\frac{14}{3} e^{-\frac{3}{4} t}$
B) $y=\frac{4}{3} e^{\frac{3}{2} t}-\frac{2}{3} e^{\frac{3}{4} t}$
C) $y=-\frac{14}{3} e^{-\frac{3}{2} t}+\frac{20}{3} e^{-\frac{3}{4} t}$
D) $y=-\frac{2}{3} e^{\frac{3}{2} t}+\frac{4}{3} e^{\frac{3}{4} t}$

Question 18

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.

Question 19

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.

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 20

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.

A) $(0,2 \pi)$
B) $(-2 \pi, 2 \pi)$
C) $(0, \infty)$
D) $\left(\frac{\pi}{12}, \frac{5 \pi}{4}\right)$
E) $\left(0, \frac{\pi}{12}\right)$
F) $\left(-\frac{5 \pi}{4}, \frac{5 \pi}{8}\right)$

Question 21

Consider the initial value problem

Which of these statements are true? Select all that apply.

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 22

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.

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 23

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

Question 24

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

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 25

Compute the Wronskian of the pair of functions

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

Question 26

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

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

Question 27

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

Question 28

Consider the pair of functions y₁ = ln t and y₁ = t ln t.
Compute the Wronskian of this function pair.

A) $\frac{1}{t^{2}}$
B) $\frac{1}{1}$
C) $\frac{\ln t}{t}$
D) $(\ln t)^{2}$
E) $\ln \left(t^{2}\right)$

Question 29

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

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 30

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

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).

Question 31

Which of these is a fundamental set of solutions for the differential equation
Select all that apply.

A) $y_{1}=\cos (10 t)$ and $y_{2}=\sin (10 t)$
B) $y_{3}=7 \cos (10 t)-20 \sin (10 t)$ and $y_{4}=10 \cos (10 t)-14 \sin (10 t)$
C) $y_{5}=e^{-10 t}$ and $y_{6}=e^{10 t}$
D) $y_{7}=e^{10 t} \sin (10 t)$ and $y_{8}=e^{10 t} \cos (10 t)$
E) $y_{9}=7 \sin (10 t)$ and $y_{10}=7 \sin (10 t)-8 \cos (10 t)$

Question 32

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

Question 33

What is the characteristic equation for the second-order homogeneous differential equation + 36y=0?

A) $(r-6)(r+6)=0$
B) $r^{2}+6=0$
C) $r^{2}+36=0$
D) $r^{2}+36 r=0$

Question 34

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

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 35

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

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 36

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

A) $y_{1}=-\frac{1}{2} \pi \sin (3 t)$
B) $y_{2}=e^{6 t} \cos (3 t)$
C) $y_{3}=2 e^{6 t}$
D) $y_{4}=5 e^{6 t}(\sin (3 t)+\cos (3 t))$
E) $y_{5}=C e^{-6 t} \cos (3 t)$ , where $C$ is any real constant
F) $y_{6}=e^{-6 t} \cos (3 t)$

Question 37

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

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 38

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

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 39

What is the solution of this initial value problem:

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 40

What is the solution of this initial value problem:

A) $y=e^{3 t}\left[3 \cos (7 t)-\frac{4}{7} \sin (7 t)\right]$
B) $y=e^{-3 t}(3 \cos (7 t)+2 \sin (7 t))$
C) $y=e^{7 t}\left(3 \cos (3 t)+\frac{14}{3} \sin (3 t)\right)$
D) $y=e^{-7 t}\left[3 \cos (3 t)-\frac{4}{3} \sin (3 t)\right]$

Question 41

Consider the initial value problem:

What is the solution of this initial value problem?

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 42

Consider the initial value problem:

Which of the following is an accurate description of the long-term behavior of the solution?

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 43

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

For any choice of $\alpha$ and $\beta$ satisfying

A) y is periodic with period $\pi$ .
B) y is periodic with period 2 $\pi$ .
C) y becomes unbounded in both the positive and negative y-directions as t $\rightarrow$ $\infty$ .
D) y oscillates toward 0 as t $\rightarrow$ $\infty$ .
E) y increases toward + $\infty$ if $\beta$ > 0, and decreases toward - $\infty$ if $\beta$ < 0.

Question 44

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

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 45

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.

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 46

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?

Question 47

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'.

Question 48

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'.

Question 49

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

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

Question 50

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

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 51

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

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 52

What is the solution of this initial value problem:

Question 53

Consider this initial value problem:

What is the solution of this initial value problem?

A) $y=\frac{14}{9} e^{-\frac{9}{2} t}+2 e^{\frac{9}{2} t}$
B) $y=\frac{22}{9} e^{-\frac{9}{2} t}-14 e^{-\frac{9}{2} t}$
C) $y=e^{\frac{9}{2} t}(2-14 t)$
D) $y=e^{-\frac{9}{2} t}(2+4 t)$

Question 54

Consider this initial value problem:

Which of the following is an accurate description of the long-term behavior of the solution?

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 55

Consider this initial value problem:

For what values of $\alpha$ does the solution tend to 0 as t $\rightarrow$ $\infty$ ?

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

Question 56

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.

Question 57

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.

Question 58

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

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 59

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.

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 60

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.

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 61

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.

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

Question 62

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.

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 63

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.

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

Question 64

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.

A) $Y(t)=A e^{8 t}\left[\sin \frac{\pi}{9} t+\cos \frac{\pi}{9} t\right]+B$
B) $Y(t)=A e^{8 t} \sin \left(\frac{\pi}{9} t\right)+B$
C) $Y(t)=e^{A t}(\sin (B t)+\cos (B t))$
D) $Y(t)=A e^{8 t} \sin \left(\frac{\pi}{9} t\right)$
E) $Y(t)=e^{8 t}\left(A \sin \frac{\pi}{9} t+B \cos \frac{\pi}{9} t\right)$

Question 65

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

A) $y(t)=C_{1} e^{-\frac{9}{2} t}+C_{2} t e^{-\frac{9}{2} t}+\mathrm{A} t^{2}$
B) $y(t)=e^{-\frac{9}{2} t}\left(A t^{2}+B t+C\right)+C_{1}$
C) $y(t)=e^{\frac{9}{2} t}\left(t+C_{1}\right)+A t^{2}+B t+C$
D) $y(t)=e^{-\frac{9}{2} t}\left(C_{1}+C_{2} t\right)+A t^{2}+B t+C$
E) $y(t)=C_{1} e^{\frac{9}{2} t}+C_{2} t e^{\frac{9}{2} t}+A t^{2}+B t$

Question 66

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.

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 67

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.

A) $Y(t)=A e^{-6 t}+B e^{9 t}+C t e^{-3 t}+D$
B) $Y(t)=(A+B t)\left(e^{-6 t}+e^{9 t}+t e^{-3 t}+C\right.$
C) $Y(t)=A e^{-6 t}+B e^{9 t}+(C t+D) e^{-3 t}+E$
D) $Y(t)=(A+B t) e^{-6 t}+(C+D t) e^{9 t}+(E+F t) e^{-3 t}+G$

Question 68

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.

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 69

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.

A) $Y(t)=A t^{2}+B t+C+D e^{-\sqrt{3} t}$
B) $Y(t)=\left(A t^{2}+B t\right) e^{-\sqrt{3} t}$
C) $Y(t)=A t(B t+3) e^{-\sqrt{3} t}$
D) $Y(t)=\left(A t^{2}+B t+C\right) e^{-\sqrt{3} t}$

Question 70

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.

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 71

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.

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 72

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.

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 73

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.

A) $Y(t)=A$
B) $Y(t)=A e^{-4 t} \sin (4 t)+B e^{-4 t} \cos (4 t)$
C) $Y(t)=A e^{4 t} \sin (-4 t)+B e^{4 t} \cos (-4 t)$
D) $Y(t)=A \sin (4 t)+B \cos (4 t)$

Question 74

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.

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

Question 75

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.

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 76

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

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 77

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.

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 78

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

A) $y(t)=C_{1}+C_{2} e^{-\frac{10}{11} t}+A e^{-\frac{10}{11} t}+(B t+C) e^{-\frac{10}{11} t}$
B) $y(t)=C_{1}+C_{2} e^{\frac{10}{11} t}+(A t+B) e^{\frac{10}{11} t}+C e^{-\frac{10}{11} t}$
C) $y(t)=C_{1} e^{-\frac{10}{11} t}+C_{2} e^{\frac{10}{11} t}+(A t+B) e^{-\frac{10}{11} t}+(C t+D) e^{\frac{10}{11} t}$
D) $y(t)=C_{1}+C_{2} e^{-\frac{10}{\pi 1} t}+A e^{-\frac{10}{11} t}+B e^{\frac{10}{11} t}$

Question 79

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

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 80

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.

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