Deck 4: Higher-Order Linear Differential Equations

Question 1

On which of these intervals is a solution of the following differential equation sure to exist? Select all that apply.-
.

A)
B) (0,
$\infty$ )
C) (- $\infty$ , 0)
D) (-1, 0)
E) (- $\infty$ , -2)F) (- $\infty$ , $\infty$ )

Answer 1

(-1, 0)
(- $\infty$ , -2)F) (- $\infty$ , $\infty$ )

Question 2

On which of these intervals is a solution of the following differential equation sure to exist? Select all that apply.
.

A) (4, 5]
B) (0, 4)
C) (0, $\infty$ )
D) (0, 5]
E) (4, $\infty$ )F) [5, $\infty$ )

Answer 2

(4, 5]
(0, 4)

Question 3

On which of these intervals is a solution of the following differential equation sure to exist? Select all that apply.-4 + - (tan t) y = -3

A) (0, $\infty$ )
B)
C) (0, $\infty$ ).
D)
E)
F)

Answer 3

Question 4

Compute the Wronskian W for the set of functions 2x, -3 , -2 .
W = ________

Question 5

Consider the fourth-order differential equation - 16y = 0. The general solution of this equation can be . are arbitrary real constants.

Question 6

Compute the Wronskian W for the set of functions , ..
W = ________

Question 7

For which of the following homogeneous differential equations is the general solution given by are arbitrary real constants?

A) $y^{\prime \prime}-16 y^{\prime}=0$
B) $y^{\prime \prime \prime}-16 y^{\prime}=0$
C) $y^{\prime \prime \prime}-16 y=0$
D) $y^{\prime \prime}-16 y=0$

Question 8

For which of the following homogeneous differential equations is the general solution given by
are arbitrary real constants?

A) $y^{(4)}+18 y^{\prime \prime}+81 y=0$
B) $y^{(4)}-18 y^{\prime \prime}+81 y=0$
C) $y^{(4)}+9 y^{\prime \prime}+81 y=0$
D) $y^{(4)}-81 y=0$
E) $y^{(4)}+81 y=0$

Question 9

For which of the following homogeneous differential equations is the general solution given by are arbitrary real constants?

A) $y^{(4)}-256 y^{\prime \prime}=0$
B) $y^{(4)}-16 y^{\prime t}=0$
C) $y^{(4)}+8 y^{\prime \prime \prime}+16 y^{\prime \prime}=0$
D) $y^{(4)}-8 y^{\prime \prime \prime}+16 y^{\prime \prime}=0$

Question 10

For which of the following homogeneous differential equations is the general solution given by are arbitrary real constants?

A) $y^{\prime \prime \prime}+-24 y^{\prime \prime}-176 y^{\prime}+-384 y=0$
B) $y^{\prime \prime \prime}--24 y^{\prime \prime}+176 y^{\prime}--384 y=0$
C) $y^{(4)}+-24 y^{\prime \prime \prime}-176 y^{\prime \prime}+-384 y^{\prime}=0$
D) $y^{(4)}--24 y^{\prime \prime \prime}+176 y^{\prime \prime}--384 y^{\prime}=0$

Question 11

What is the least order of a homogeneous differential equation with constant coefficients for which is a solution?

Question 12

What is the least order of a homogeneous differential equation with constant coefficients for which is a solution?

Question 13

What is the least order of a homogeneous differential equation with constant coefficients for which is a solution?

Question 14

What is the least order of a homogeneous differential equation with constant coefficients for which is a solution?

Question 15

What is the least order of a homogeneous differential equation with constant coefficients for which is a solution?

Question 16

What is the least order of a homogeneous differential equation with constant coefficients for which is a solution?

Question 17

What is the least order of a homogeneous differential equation with constant coefficients for which is a solution?

Question 18

What is the least order of a homogeneous differential equation with constant coefficients for which is a solution?

Question 19

For which of the following homogeneous differential equations is the general solution given by
are arbitrary real constants?

A) $y^{(4)}-81 y=0$
B) $y^{(4)}+81 y=0$
C) $y^{(4)}-81 y^{\prime}=0$
D) $y^{(4)}+81 y^{\prime}=0$

Question 20

What is the general solution of the third-order homogeneous differential equation

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

Question 21

What is the general solution of the third-order homogeneous differential equation ?

A) $y=C_{1}+C_{2} e^{2 t} \cos (4 t)+C_{3} e^{2 t} \sin (4 t)$
B) $y=C_{1}+C_{2} e^{2 t} \cos (2 t)+C_{3} e^{2 t} \sin (2 t)$
C) $y=C_{1}+C_{2} e^{-2 t} \cos (2 t)+C_{3} e^{-2 t} \sin (2 t)$
D) $y=C_{1}+C_{2} e^{4 t} \cos (2 t)+C_{3} e^{4 t_{5} \sin (2 t)}$
E) $y=C_{1}+C_{2} e^{-4 t} \cos (2 t)+C_{3} e^{-4 t} \sin (2 t)$

Question 22

What is the general solution of the fifth-order homogeneous differential equation

A) $y=C_{1}+C_{2} t+C_{3} t^{2}+C_{4} e^{3 t}+C_{5} t e^{3 t}$
B) $y=C_{1}+C_{2} t+C_{3} t^{2}+C_{4} e^{-3 t}+C_{5} t e^{-3 t}$
C) $y=C_{1}+C_{2} t+C_{3} e^{3 t}+C_{4} t e^{3 t}+C_{5} t^{2} e^{3 t}$
D) $y=C_{1}+C_{2} t+C_{3} e^{-3 t}+C_{4} t e^{-3 t}+C_{5} t^{2} e^{-3 t}$

Question 23

What is the general solution of the third-order homogeneous differential equation

A) $y=C_{1}+C_{2} e^{2 t} \cos \left(\frac{\sqrt{32}}{2} t\right)+C_{3} e^{2 t} \cos \left(\frac{\sqrt{32}}{2} t\right)$
B) $y=C_{1}+C_{2} e^{-2 t} \cos \left(\frac{\sqrt{32}}{2} t\right)+C_{3} e^{-2 t} \cos \left(\frac{\sqrt{32}}{2} t\right)$
C) $y=C_{1}+C_{2} e^{\frac{4-\sqrt{32}}{2} t}+C_{3} e^{\frac{4+\sqrt{32}}{2} t}$
D) $y=C_{1}+C_{2} e^{\frac{-4-\sqrt{32}}{2}}+C_{3} e^{\frac{-4+\sqrt{32}}{2}}$

Question 24

The differential equation EI⁽⁴⁾ - ky = 0, where E, I, and k are positive constants, arises when modeling vibrating beams. What is the general solution of this equation?

A)
B)
C)
D)

Question 25

Suppose a fourth-order homogeneous linear differential equation with constant coefficients has these characteristic roots: -8.8, 8.8, 10i, -10i
Which of the following terms are part of the general solution of this equation? Here, C denotes an arbitrary real constant.
Select all that apply.

A) $C$
B) $\mathrm{Ce}^{-8.8 \mathrm{t}}$
C) $\mathrm{Ct}^{-8.81}$
D) $C e^{-8.8 t} \sin (10 t)$
E) $\cos (10 t)+C$
F) $C \sin (10 t)$

Question 26

Suppose a seventh-order homogeneous linear differential equation with constant coefficients has these characteristic roots: 0, 0, 0, 3 - i, 3 + i, - i, i
Which of the following terms are part of the general solution of this equation? Here, C denotes an arbitrary real constant.Select all that apply.

A) $\mathrm{Ct}$
B) $\mathrm{Ct}^{3}$
C) $t^{2}+C$
D) $\mathrm{Ce}^{3 t} \cos (\sqrt{3} t)$
E) $\mathrm{Ce}^{3 t}$
F) $C e^{\sqrt{3} t} \sin (3 t)$
G) $t e^{\sqrt{3} t}+C$
H) $C \sin (\sqrt{3} t)$

Question 27

Suppose a ninth-order homogeneous linear differential equation with constant coefficients has these characteristic roots: -6i, -6i, 6i, 6i, , - , - , -6.2
Which of the following terms are part of the general solution of this equation? Here, C denotes an arbitrary real constant.Select all that apply.

A) $C t \sin (6 t)$
B) $\mathrm{Ct}$
C) $\mathrm{C}$
D) $C \sqrt{11}$
E) $C t^{2} e^{-\sqrt{11} t}$
F) $C e^{\sqrt{11} t} \cos (6 t)$
G) $C t^{2} \cos (6 t)$
H) $\mathrm{Ce}-\sqrt{11} t$

Question 28

What is the general solution of the third-order linear homogeneous differential equation

Question 29

What is the general solution of the third-order linear homogeneous differential equation

Question 30

What is the general solution of the third-order linear homogeneous differential equation ?

Question 31

What is the general solution of the fourth-order linear homogeneous differential equation

Question 32

Solve the following initial value problem:

Question 33

Solve the following initial value problem:

Question 34

Solve the following initial value problem:

Question 35

What is the form of a particular solution of the following nonhomogeneous differential equation?

All capital letters in the choices represent arbitrary real constants.

A) $A e^{9 t}+B e^{64 t}$
B) $(A t+B) e^{9 t}+(C t+D) e^{64 t}$
C) $A\left(e^{9 t}+e^{64 t}\right)$
D) $A t\left(e^{9 t}+e^{64 t}\right)$

Question 36

What is the form of a particular solution of the following nonhomogeneous differential equation?

All capital letters in the choices represent arbitrary real constants.

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

Question 37

What is the form of a particular solution of the following nonhomogeneous differential equation?

All capital letters in the choices represent arbitrary real constants.

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

Question 38

What is the form of a particular solution of the following nonhomogeneous differential equation?

All capital letters in the choices represent arbitrary real constants.

A) $(A t+B) e^{5 t}+(C t+D) e^{10 t}$
B) $A e^{5 t} \sin (5 t)+B e^{5 t} \cos (5 t)+C e^{10 t}$
C) $e^{5 t}(A \sin (5 t)+B \cos (5 t))+C e^{10 t}+D t$
D) $A \sin (5 t)+B \cos (5 t)+(C t+D) e^{10 t}$

Question 39

What is the form of a particular solution of the following nonhomogeneous differential equation?

All capital letters in the choices represent arbitrary real constants.

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

Question 40

What is the form of a particular solution of the following nonhomogeneous differential equation?

All capital letters in the choices represent arbitrary real constants.

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

Question 41

What is the form of a particular solution of the following nonhomogeneous differential equation?-

All capital letters in the choices represent arbitrary real constants.

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

Question 42

What is the form of a particular solution of the following nonhomogeneous differential equation?

All capital letters in the choices represent arbitrary real constants.

A) $(A t+B) e^{2 t} \sin (2 t)$
B) $A e^{2 t} \sin (2 t)+B e^{2 t} \cos (2 t)$
C) $A e^{2 t} \sin (2 t)$
D) $(A t+B) e^{2 t} \sin (2 t)+(C t+D) e^{2 t} \cos (2 t)$
E) $(A t+B) e^{2 t}(\sin (2 t)+\cos (2 t))$

Question 43

What is the form of a particular solution of the following nonhomogeneous differential equation?

All capital letters in the choices represent arbitrary real constants.

A) $A e^{-5 t}$
B) $(A t+B) e^{-5 t}$
C) $\left(A t^{2}+B t+C\right) e^{-5 t}$
D) $\left(A t^{3}+B t^{2}+C t+D\right) e^{-5 t}$

Question 44

What is the form of a particular solution of the following nonhomogeneous differential equation?

Use capital letters A, B, C, ... to represent arbitrary real constants. Do not actually solve for the constants.

Question 45

What is the form of a particular solution of the following nonhomogeneous differential equation?

Use capital letters A, B, C, ... to represent arbitrary real constants. Do not actually solve for the constants.

Question 46

Consider the nonhomogeneous differential equation The functions form a fundamental set of solutions of the corresponding homogeneous equation
Compute the Wronskian,

Question 47

Consider the nonhomogeneous differential equation The functions form a fundamental set of solutions of the corresponding homogeneous equation
Suppose the method of variation of parameters is to be used to find a particular solution Y_p of the nonhomogeneous differential equation. Then, for appropriately-chosen functions As part of the process of forming this solution, find the following quantities.

Question 48

Consider the following third-order nonhomogeneous differential equation:

Find three functions Y₁ , Y₁ and Y₁ that form a fundamental set and are such that the general solution of the corresponding homogeneous equation is given by are arbitrary real constants.
Y₁ = ________, Y₂ = ________, Y₃ = ________

Question 49

Consider the following third-order nonhomogeneous differential equation:

Compute the Wronskian, W(t), for the set of functions Y₁ ,Y₂ , and Y₃ that form a fundamental set and are such that the general solution of the corresponding homogeneous equation + 36 are arbitrary real constants.

Question 50

Consider the following third-order nonhomogeneous differential equation:

Suppose the method of variation of parameters is to be used to find a particular solution of the nonhomogeneous differential equation. Then, for appropriately-chosen functions u₁ , u₂, and u₃ . As part of the process of forming this solution, find the following quantities.

Question 51

Use the method of variation of parameters to solve the following third-order nonhomogeneous differential equation:

A) $y=C_{1} e^{t}+C_{2} t e^{t}+C_{3} t^{2} e^{t}-\frac{2}{5} t^{\frac{5}{2}} e^{t}$
B) $y=C_{1} e^{t}+C_{2} t e^{t}+C_{3} t^{2} e^{t}+\frac{8}{105} t^{\frac{7}{2}} e^{t}$
C) $y=C_{1} e^{t}+C_{2} t e^{t}+C_{3} t^{2} e^{t}+\left(\frac{1}{7} t^{\frac{7}{2}}-\frac{2}{5} t^{\frac{7}{2}}\right) e^{t}$
D) $y=C_{1} e^{-t}+C_{2} t e^{-t}+C_{3} t^{2} e^{-t}+\frac{8}{105} t^{\frac{7}{2}} e^{t}$
E) $y=C_{1} e^{-t}+C_{2} t e^{-t}+C_{3} t^{2} e^{-t}+\left(\frac{1}{7} t^{\frac{7}{2}}-\frac{2}{5} t^{\frac{7}{2}}\right) e^{t}$
F) $y=C_{1} e^{-t}+C_{2} t e^{-t}+C_{3} t^{2} e^{-t}-\frac{2}{5} t^{\frac{5}{2}} e^{t}$

Question 52

Compute the Wronskian W for the set of functions

Question 53

What is the general solution of the third-order homogeneous Cauchy Euler differential equation

A) $y=C_{1} x+C_{2} x \ln x+C_{3} x^{2} \ln x$
B) $y=C_{1} x+C_{2} x \ln x+C_{3} x(\ln x)^{2}$
C) $y=C_{1} x^{-1}+C_{2} x^{-1} \ln \mathrm{x}+C_{3} x^{-1}(\ln x)^{2}$
D) $y=C_{1} x^{-1}+C_{2} \ln \left(x^{-1}\right)+C_{3}\left(\ln \left(x^{-1}\right)\right)^{2}$

Question 54

What is the general solution of the third-order homogeneous Cauchy Euler differential equation

A) $y=C_{1} \ln x+C_{2} \ln \left(x^{-1}\right)+C_{3} \ln \left(x^{2}\right)$
B) $y=C_{1} x^{-2}+C_{2} x^{-1}+C_{3} x$
C) $y=C_{1} x^{2}+C_{2} x+C_{3} x^{-1}$
D) $y=C_{1} \ln x+C_{2}(\ln x)^{2}+C_{3}(\ln x)^{-1}$