Exam Questions + Second-Order Linear Differential + Ch3 - Complete Test Bank | Differential Equations 12e by William E. Boyce. DOCX document preview.
Elementary Differential Equations, 12e (Boyce)
Chapter 3 Second-Order Linear Differential Equations
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
Type: MC Var: 1
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
Type: MC Var: 1
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
Type: MC Var: 1
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
Type: MC Var: 1
5) Which of the following are solutions to the homogeneous second-order differential equation 
? Select all that apply.
A) 
= 8
+ 2
B) 
= C, where C is any real constant
C) 
= 8
D) 
= C, where C is any real constant
E) 
= 
∙ 
, where 
and 
are any real constants
F) 
= 2
G) 
= C 
, where C is any real constant
Type: MC Var: 1
6) Which of the following are solutions to the homogeneous second-order differential equation 
? Select all that apply.
A) = 
, where C is any real constant
B) = -4
+ 3
C) = C
, where C is any real constant
D) = C
, where C is any real constant
E) = 3 + -4
F) = t
Type: MC Var: 1
7) Which of the following is the general solution of the homogeneous second-order differential equation + 15 + 50y = 0? Here, C, , and are arbitrary real constants.
A) y = C(
+ 
)
B) y = 
+ 
C) y = +
D) y = C( + )
E) y = + + y + 
∙ 
F) y = ∙
Type: MC Var: 1
8) Which of the following is the general solution of the homogeneous second-order differential equation 4 + 24 = 0? Here, C, , and are arbitrary real constants.
A) y = 4 + C
B) y = 
+
C) y = +
D) y = + 
E) y = 4 + C
Type: MC Var: 1
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) - 
B) -4
C) 0
D)
E) 4
Type: MC Var: 1
10) What is the solution of the initial value problem
- 3 = 0, y(0) = 3, (0) = -2?
A) y = - 
- 
B) y = -2t + 
C) y = - 
D) y = - +
Type: MC Var: 1
11) Consider the second-order homogeneous differential equation - 12 + 32y = 0.
What is the general solution of this differential equation? Here, C, , and are arbitrary real constants.
A) y = + 
B) y = 
+ 
C) y = 
+ 
D) y = 
+ 
E) y = 
+
Type: MC Var: 1
12) Consider the second-order homogeneous differential equation - 6 + 8y = 0.
If the differential equation is equipped with the initial conditions y(0) = 6, (0) = 6, what is the solution of the resulting initial value problem?
A) y = 9 - 3
B) y = 6 + 6
C) y = -9
- 3
D) y = 6 + 6
Type: MC Var: 1
13) Consider the initial value problem
+ 8 = 0, y(0) = -4, (0) = 16
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
Type: MC Var: 1
14) Consider the initial value problem
+ 8 = 0, y(0) = 2, (0) = 24
Fill in the blank: 
y(t) = ________
Type: SA Var: 1
15) Consider the initial value problem
49 - 16y = 0, y(0) = α, (0) = 4
What is the solution of this initial value problem?
A) y = 
+ 
B) y = +
C) y = 
+ 
D) y = +
Type: MC Var: 1
16) Consider the initial value problem
36 - 9y = 0, y(0) = α, (0) = 3
For what value of α does the solution of this initial value problem tend to zero as t → ∞?
A) - 6
B) - 
C) 0
D) 6
E)
Type: MC Var: 1
17) Consider the initial value problem
8 + 18 + 9y = 0, y(0) = 2, (0) = 2
What is the solution of this initial value problem?
A) y = 
- 
B) y = 
-
C) y = - +
D) y = - +
Type: MC Var: 1
18) Consider the initial value problem
8 + 30 + 25y = 0, y(0) = -2, (0) = 8
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.
Type: SA Var: 1
19) Consider the initial value problem
(8
- 9t + 1) - 8ty = sin(2πt), y
= -4, = -3.5
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) 
C) (0, 1)
D) 
E) (-∞, ∞)
Type: MC Var: 1
20) Consider the initial value problem
(4 + ) + tan
+ 3 ln |t| y = -4t, y
= 3, = -3
On which of these intervals is this initial value problem certain to have a unique twice differentiable solution? Select all that apply.
A) (0, 2π)
B) (-2π, 2π)
C) (0, ∞)
D) 
E) 
F) 
Type: MC Var: 1
21) Consider the initial value problem
+ cos
+ y = 0, y(-1) = 0, (-1) = 0
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 = φ(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 
.
Type: MC Var: 1
22) Suppose that and are both solutions of the differential equation 
. Which of the following must also be solutions of this differential equation? Select all that apply. Here, , and are arbitrary real constants.
A) 5 - 4
B) t
C)
D) () ∙ ()
E) ( + )
F) (7 - 9) - (2 - 7)
Type: MC Var: 1
23) If and are both solutions of the differential equation + 2
- 5ty = 3, then - is also a solution of this equation.
Type: TF Var: 1
24) Consider the differential equation - 5t + ( - 1)y = 1. Which of the following statements is true?
A) If 2 is a solution of this differential equation, then so is .
B) If and are both solutions of this differential equation, then - cannot be a solution of it.
C) The Principle of Superposition guarantees that if and are both solutions of this differential equation, then + must also be a solution of it, for any choice of real constants and .
D) There exist nonzero real constants and such that - is a solution of this differential equation.
Type: MC Var: 1
25) Compute the Wronskian of the pair of functions and .
A) -2
B) -6
C) -8
D) -6
E) -8
Type: MC Var: 1
26) Compute the Wronskian of the pair of functions sin(5t) and cos(5t).
A) -5
B) -4
C) 1
D) 4
E) 5
Type: MC Var: 1
27) Compute the Wronskian of the pair of functions 2t and 4.
Type: SA Var: 1
28) Consider the pair of functions = ln t and = t ln t.
Compute the Wronskian of this function pair.
A) 
B) 
C) 
D) 
E) ln()
Type: MC Var: 1
29) Consider the pair of functions = ln t and = t ln t.
Which of these statements is true?
A) Both and can be solutions of the differential equation + p(t) + q(t) y = 0 on the interval (0, ∞), where p(t) and q(t) are continuous on (0, ∞).
B) The Wronskian for this function pair is strictly positive on (0, ∞).
C) Abel's theorem implies that and cannot both be solutions of any differential equation of the form + p(t) + q(t) y = 0 on the interval (0, ∞).
D) The pair and constitutes a fundamental set of solutions to some second-order differential equation of the form + p(t) + q(t) y = 0 on the interval (0, ∞).
Type: MC Var: 1
30) Consider the pair of functions = t and = 3.
Which of these statements are true? Select all that apply.
A) W[, ](t) > 0 for all values of t in the interval (-2, 2).
B) W[, ](t) = 3
C) The pair and constitutes a fundamental set of solutions to some second-order differential equation of the form + p(t) + q(t) y = 0 on the interval (-2, 2).
D) Abel's theorem implies that and cannot both be solutions of any differential equation of the form + p(t) + q(t) y = 0 on the interval (-2, 2).
E) Since there exists a value of 
in the interval (-2, 2) for which W[, ](t) = 0, there must exist a differential equation of the form + p(t) + q(t) y = 0 for which the pair and constitute a fundamental set of solutions on the interval (-2, 2).
Type: MC Var: 1
31) Which of these is a fundamental set of solutions for the differential equation + 100y = 0? Select all that apply.
A) = cos(10t) and = sin(10t)
B) = 7 cos(10t) - 20 sin(10t) and = 10 cos(10t) - 14 sin(10t)
C) = and =
D) = sin(10t) and 
= cos(10t)
E) 
= 7 sin(10t) and 
= 7 sin(10t) - 8 cos(10t)
Type: MC Var: 1
32) The pair of functions = and = t forms a fundamental set of solutions for the differential equation - 12 + 36y = 0.
Type: TF Var: 1
33) What is the characteristic equation for the second-order homogeneous differential equation 
?
A) (r - 6)(r + 6) = 0
B) + 6 = 0
C) + 36 = 0
D) + 36r = 0
Type: MC Var: 1
34) For which of these differential equations is the characteristic equation given by 
?
A) + 50 = 0
B) + 50y = 0
C) - 2 + 50 = 0
D) - 2 + 50y = 0
E) ( - (1 -7i))( - (1 + 7i)) = 0
F) ( - (1 - 7i)y)( - (1 + 7i)y) = 0
Type: MC Var: 1
35) Which of the following are solutions to the homogeneous second-order differential equation 
? Select all that apply.
A) = 2 sin
B) = C 
, where C is any real constant
C) = -2 cos
D) = 
E) = + 
, where and are any real constants
F) = 5
+ 7
G) = sin + C, where C is any real constant
Type: MC Var: 1
36) Which of the following are solutions to the homogeneous second-order differential equation 
? Select all that apply.
A) = - 
π sin(3t)
B) = cos(3t)
C) = 2
D) = 5(sin(3t) + cos(3t))
E) = Ccos(3t), where C is any real constant
F) = 
cos(3t)
G) = sin(3t) + cos(3t), where and are any real constants
Type: MC Var: 1
37) Which of the following is the general solution of the homogeneous second-order differential equation 9 + y = 0? Here, C, , and are arbitrary real constants.
A) y = C 
B) y = cos(3t) + sin(3t)
C) y = C(cos(3t) + sin(3t))
D) y = cos 
+ sin
E) y = cos + sin + C
F) y = cos(3t) + sin(3t) + C
Type: MC Var: 1
38) Which of the following is the general solution of the homogeneous second-order differential equation + 8 + 52y = 0? Here, C, , and are arbitrary real constants.
A) y = sin(6t) + cos(6t)
B) y = (sin(6t) + cos(6t))
C) y = cos(6t) + sin(6t) + C
D) y = (sin(4t) + cos(6t)) + C
E) y = sin(6t) + cos(6t) + C
Type: MC Var: 1
39) What is the solution of this initial value problem:
+ 121y = 0, y(4π) = 3, (4π) = 10
A) y = 3cos(11t) + 
sin(11t)
B) y = cos(11t) + sin(11t)
C) y = 3 sin(11t) + 10cos(11t)
D) y = 10
+ 3
E) y = 3
+ 
Type: MC Var: 1
40) What is the solution of this initial value problem:
+ 6 + 58y = 0, y(0) = 3, (0) = 5
A) y = 
B) y = 
C) y = 
D) y = 
Type: MC Var: 1
41) Consider the initial value problem:
36 + y = 0, y(21π) = 2, (21π) = -2
What is the solution of this initial value problem?
A) y = 2 sin
+ 12 cos
B) y = -12 cos - 2 sin
C) y = 2 cos(6t) - 2 sin(6t)
D) y = -2 cos(6t) - 2sin(6t)
Type: MC Var: 1
42) Consider the initial value problem:
100 + y = 0, y(- 25π) = 4, (- 25π) = -3
Which of the following is an accurate description of the long-term behavior of the solution?
A) y(t) decreases to 0 as t → ∞.
B) y(t) is periodic with period 20π.
C) y(t) oscillates toward 0 as t → ∞.
D) y(t) becomes unbounded in both the positive and negative y-directions as t → ∞.
Type: MC Var: 1
43) Which of the following is an accurate description of the long-term behavior of the solution 
of the initial value problem
+ 2 + 26y = 0, y(0) = α, (0) = β
for any choice of α and β satisfying 
+ 
≠ 0?
A) y is periodic with period 
π.
B) y is periodic with period 2π.
C) y becomes unbounded in both the positive and negative y-directions as t → ∞.
D) y oscillates toward 0 as t → ∞.
E) y increases toward +∞ if β > 0, and decreases toward -∞ if β < 0.
Type: MC Var: 1
44) Which of the following are solutions to the homogeneous second-order Cauchy Euler differential equation + 12t - 12y = 0, t > 0? Select all that apply.
A) y = 
+ t
B) y = C
, where C is any real constant
C) y = 16t
D) y = C 
, where C is any real constant
E) y = -9 + C, where C is any real constant
F) y = + t + C, where C, , and are arbitrary real constants
Type: MC Var: 1
45) Consider the homogeneous second-order Cauchy Euler differential equation
+ 5t - 60y = 0, t > 0
What is the general solution of this differential equation? Here, and are arbitrary real constants.
A) y = 
+ 
B) y = 
+ 
C) y = 
+ 
D) y = 
+ 
E) y = 
+
Type: MC Var: 1
46) Consider the homogeneous second-order Cauchy Euler differential equation
+ 4t - 40y = 0, t > 0
What is the solution of the initial value problem comprised of this differential equation and the initial conditions y(1) = α, (1) = 4?
Type: SA Var: 1
47) Consider the homogeneous second-order Cauchy Euler differential equation
+ 4t - 40y = 0, t > 0
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'.
Type: SA Var: 1
48) Consider the homogeneous second-order Cauchy Euler differential equation
+ 4t - 40y = 0, t > 0
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'.
Type: SA Var: 1
49) For what value(s) of α is y = 
a solution of the second-order homogeneous differential equation 4 - 4 + 1y = 0?
A)
B) 0 and
C) 0 and -
D) -
E) - and
Type: MC Var: 1
50) Which of the following are solutions to the homogeneous second-order differential equation 
? Select all that apply.
A) = +
B) = -6t + 8
C) = + t, where and are arbitrary real constants
D) = 8
E) = C + 10t
F) = 2 + 8t + 8
Type: MC Var: 1
51) Which of the following is the general solution of the homogeneous second-order differential equation 16 + 24 + 9y = 0? Here, and are arbitrary real constants.
A) y = + t
B) y = +
C) y = + t
D) y = + t
E) y = t +
Type: MC Var: 1
52) What is the solution of this initial value problem:
9 + -48 + 64y = 0, y(0) = -2, (0) = 1
Type: SA Var: 1
53) Consider this initial value problem:
4 + 36 + 81y = 0, y(0) = 2, (0) = -5
What is the solution of this initial value problem?
A) y = 
+ 2
B) y = 
- 14
C) y = (2 - 14t)
D) y = (2 + 4t)
Type: MC Var: 1
54) Consider this initial value problem:
4 + 36 + 81y = 0, y(0) = 4, (0) = 7
Which of the following is an accurate description of the long-term behavior of the solution?
A) y(t) tends to 0 as t → ∞.
B) y(t) is strictly increasing and approaches ∞ as t → ∞.
C) y(t) is strictly decreasing and approaches -∞ as t → ∞.
D) y(t) becomes unbounded in both the positive and negative y-direction as t → ∞.
Type: MC Var: 1
55) Consider this initial value problem:
9 - 6α + y = 0, y(0) = 2, (0) = 1
For what values of α does the solution tend to 0 as t → ∞?
A) all real numbers
B) all nonzero real numbers
C) all positive real numbers
D) all negative real numbers
Type: MC Var: 1
56) Use the method of reduction of order to find a second solution of the differential equation 
, t > 0, using the fact that = 
is a solution.
Type: SA Var: 1
57) Use the method of reduction of order to find a second solution of the differential equation 
, t > 0, using the fact that = t is a solution.
Type: SA Var: 1
58) What is the general solution of the homogeneous second-order Cauchy Euler differential equation + 13t + 36y = 0, t > 0. Here, and are arbitrary real constants.
A) y = 
+
B) y = 
+ 
C) y = ( + ln t)
D) y = +
Type: MC Var: 1
59) Consider this second-order nonhomogeneous differential equation:
+ 4 - 21y = 4t + 7
Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, and are arbitrary real constants.
A) y(t) = +
B) y(t) = + t
C) y(t) = + t
D) y(t) = +
E) y(t) = + t
Type: MC Var: 1
60) Consider this second-order nonhomogeneous differential equation:
+ -4 - 12y = -2t + 1
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) = ( + ) ∙ (At + B)
B) Y(t) = At + B
C) Y(t) = At
D) Y(t) = At + +
Type: MC Var: 1
61) Consider this second-order nonhomogeneous differential equation:
+ 7 + 10y = 
Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, and are arbitrary real constants.
A) y(t) = + +
B) y(t) = ( + ) +
C) y(t) = +
D) y(t) = +
E) y(t) = ( + ) +
Type: MC Var: 1
62) Consider this second-order nonhomogeneous differential equation:
+ 11 + 28y = 
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
B) Y(t) = 
C) Y(t) = A
D) Y(t) = A +
E) Y(t) = A + B
Type: MC Var: 1
63) Consider this second-order nonhomogeneous differential equation:
+ 16 + 60y = 6sin
Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, and are arbitrary real constants.
A) y(t) = ( + ) +
B) y(t) = +
C) y(t) = +
D) y(t) = +
E) y(t) = + ( + t)
Type: MC Var: 1
64) Consider this second-order nonhomogeneous differential equation:
+ 11 + 28y = 7sin
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
+ B
B) Y(t) = Asin + B
C) Y(t) = (sin(Bt) + cos(Bt))
D) Y(t) = Asin
E) Y(t) = 
Type: MC Var: 1
65) Which of these is the general solution of the second-order nonhomogeneous differential equation 4 + 36 + 81y = -9 - 7
? Here, , and , and all capital letters are arbitrary real constants.
A) y(t) = + t + A
B) y(t) = (A + Bt + C) +
C) y(t) = (t + ) + A + Bt + C
D) y(t) = ( + t) + A + Bt + C
E) y(t) = + t + A + Bt
Type: MC Var: 1
66) Consider this second-order nonhomogeneous differential equation:
25 - 20 + 4y = + + t - 17
Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, and are arbitrary real constants.
A) y(t) = 
( + t)
B) y(t) = + 
C) y(t) = (t + ) +
D) y(t) = ( + t)
E) y(t) = (t + ) +
Type: MC Var: 1
67) Consider this second-order nonhomogeneous differential equation:
49 - 56 + 16y = + 
+ t + 9
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 + Ct
+ D
B) Y(t) = (A + Bt)( + + t) + C
C) Y(t) = A + B + (Ct + D) + E
D) Y(t) = (A + Bt) + (C + Dt) + (E + Ft) + G
Type: MC Var: 1
68) Consider this second-order nonhomogeneous differential equation:
+ 25y = 
t(-8 + 4t)
Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, and are arbitrary real constants.
A) y(t) = sin(5t) + cos(5t)
B) y(t) = sin(25t) + cos(25t)
C) y(t) = +
D) y(t) = +
Type: MC Var: 1
69) Consider this second-order nonhomogeneous differential equation:
+ 4y = t(-7 + 8t)
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 + Bt + C + D
B) Y(t) = (A + Bt)
C) Y(t) = At(Bt + 3)
D) Y(t) = (A + Bt + C)
Type: MC Var: 1
70) Consider this second-order nonhomogeneous differential equation:
- 4 + 20y = -4t - 21
Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, and are arbitrary real constants.
A) y(t) = (sin(4t) + cos(4t)) +
B) y(t) = sin(2t) + cos(2t)
C) y(t) = (sin(2t) + cos(2t)) +
D) y(t) = sin(4t) + cos(4t)
Type: MC Var: 1
71) Consider this second-order nonhomogeneous differential equation:
- 8 + 20y = 13t - 9
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) = At + B
B) Y(t) = (At + B) sin(2t) + (Ct + D) cos(2t)
C) Y(t) = A + B + C + Dt + E
D) Y(t) = (A + B + C + Dt + E)(sin(2t) + cos(2t))
E) Y(t) = (A + Bt)sin(4t) +(C + Dt)cos(4t)
F) Y(t) = (A + B + C + Dt + E)(sin(4t) + cos(4t))
Type: MC Var: 1
72) Consider this second-order nonhomogeneous differential equation:
+ 10 + 50y = -8π
Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, and are arbitrary real constants.
A) y(t) = (sin(-5t) + cos(-5t)) +
B) y(t) = sin(5t) + cos(5t)
C) y(t) = sin(-5t) + cos(-5t)
D) y(t) = (sin(5t) + cos(5t)) +
Type: MC Var: 1
73) Consider this second-order nonhomogeneous differential equation:
+ 8 + 32y = -3π
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) = Asin(4t) + Bcos(4t)
C) Y(t) = Asin(-4t) + Bcos(-4t)
D) Y(t) = A sin(4t) + B cos(4t)
Type: MC Var: 1
74) Consider this second-order nonhomogeneous differential equation:
36 + 25 = - 
t cos(3t)
Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, and are arbitrary real constants.
A) y(t) = (sin(5t) + cos(5t))
B) y(t) = 
+ 
C) y(t) = 
+ 
D) y(t) = sin
+ cos
E) y(t) = sin
+ cos
Type: MC Var: 1
75) Consider this second-order nonhomogeneous differential equation:
9 + 4 = - 
t cos(3t)
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) = At cos(3t)
B) Y(t) = At cos(3t) + Bt sin(3t)
C) Y(t) = (At + B)sin(3t) + (Ct + D)cos(3t)
D) Y(t) = (A + Bt + C)sin(3t) + (D + Et + F)cos(3t)
E) Y(t) = Acos(3t) + Bsin(3t)
Type: MC Var: 1
76) Consider this second-order nonhomogeneous differential equation:
36 + y = 
sin(6t) + 6 + ln(2)
Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, and are arbitrary real constants.
A) y(t) = sin(6t) + cos(6t)
B) y(t) = sin + cos
C) y(t) = 
+ 
D) y(t) = +
E) y(t) = 
+
Type: MC Var: 1
77) Consider this second-order nonhomogeneous differential equation:
25 + y = 
sin(5t) + 7 + ln(3)
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(5t) + B + C
B) Y(t) = (At + B)sin(5t) + C + D
C) Y(t) = (A sin(5t) + B cos(5t) + C) + D
D) Y(t) = A sin(5t) + B cos(5t) + C + D
E) Y(t) = (At + B)sin(5t) + (Ct + D)cos(5t) + E + F
Type: MC Var: 1
78) Which of these is the general solution of the second-order nonhomogeneous differential equation 11 + 10 = 6
+ 4
? Here, , , and all capital letters are arbitrary real constants.
A) y(t) = + + A + (Bt + C)
B) y(t) = + + (At + B) + C
C) y(t) = + + (At + B) + (Ct + D)
D) y(t) = + + A + B
Type: MC Var: 1
79) Which of these is the general solution of the second-order nonhomogeneous differential equation 7 = 
sin(
t) - 3 cos
? Here, , , and all capital letters are arbitrary real constants.
A) y(t) = + t + A sin( t) + B cos( t) + C sin + D cos
B) y(t) = + t + A sin( t) + B cos( t) + C sin + D cos
C) y(t) = + t + A sin( t) + B cos
D) y(t) = t + A sin( t) + B cos
E) y(t) = t + A sin( t) + B cos( t) + C sin + D cos
Type: MC Var: 1
80) Consider this second-order nonhomogeneous differential equation:
- 6 + 25y = -6 + 9 cos(4t)
Which of the following is the form of the solution of the corresponding homogeneous differential equation? Here, and are arbitrary real constants.
A) y(t) = (sin(4t) + cos(4t)) +
B) y(t) = (sin(3t) + (cos(3t)
C) y(t) = (sin(3t) + cos(3t)) +
D) y(t) = (sin(4t) + (cos(4t)
Type: MC Var: 1
81) Consider this second-order nonhomogeneous differential equation:
- 10 + 74y = -6 + 8 cos(7t)
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 cos(7t)
B) Y(t) = (At + B) + (Ct + D)cos(7t)
C) Y(t) = A + B cos(7t) + C sin(7t)
D) Y(t) = (At + B)cos(7t) + (Ct + D)sin(7t)
Type: MC Var: 1
82) Consider this second-order nonhomogeneous differential equation:
- 25y = 6
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.
Type: SA Var: 1
83) Consider this second-order nonhomogeneous differential equation:
3 - 8 = -15 + 3t + 
t
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.
Type: SA Var: 1
84) Which of these is the general solution of the second-order nonhomogeneous differential equation 4 + 15 - 4y = -5 + 3 + 9? Here, 
, 
, and all capital letters are arbitrary real constants.
A) y(t) = + 
+ A + B + C
B) y(t) = + + (At + B) + C + D
C) y(t) = + 
+ A + B + C
D) y(t) = + + (At + B) + C + D
Type: MC Var: 1
85) Which of these is the general solution of the second-order nonhomogeneous differential equation 5 + 7 = t 
? Here, , , and all capital letters are arbitrary real constants.
A) y(t) = + + At + B + (Ct + D)
B) y(t) = + + A + Bt + C + (D + Et + F)
C) y(t) = + + 
D) y(t) = + + A + Bt + C + (D + Et + F)
E) y(t) = + +
Type: MC Var: 1
86) Which of these is the general solution of the second-order nonhomogeneous differential equation 4 - 4 + 1y = 
+ 14
? Here, , , and all capital letters are arbitrary real constants.
A) y(t) = + t + (A + Bt + C) + D
B) y(t) = + t + A + B
C) y(t) = ( + t) + (At + B) + C
D) y(t) = 
( + t) + (A + Bt + C) + D
E) y(t) = ( + t) + A + B
Type: MC Var: 1
87) Which of these is the general solution of the second-order nonhomogeneous differential equation + 64y = 
cos(8t)? Here, , , and all capital letters are arbitrary real constants.
A) y(t) = cos(8t) + sin(8t) + (A + Bt + C)cos(8t)
B) y(t) = sin(8t) + cos(8t) + (A + Bt + C)sin(8t) + (D + Et + F)cos(8t)
C) y(t) = sin(8t) + cos(8t) + (At + B)sin(8t) + (Ct + D)cos(8t)
D) y(t) = + + (At + B)sin(8t) + (Ct + D)cos(8t)
E) y(t) = + + (At + B)cos(8t)
Type: MC Var: 1
88) Which of these is the general solution of the second-order nonhomogeneous differential equation 9 + 1y = cos
+ 5t sin? Here, , , and all capital letters are arbitrary real constants.
A) y(t) = cos
+ + A cos + (Bt + C)sin
B) y(t) = cos + + (At + B)cos + (C + Dt + E)sin
C) y(t) = cos + sin + (At + B) cos + (B + Dt + E)sin
D) y(t) = cos + sin + A cos + (Bt + C)sin
E) y(t) = cos + sin + (A + Bt + C)cos + (D + Et + F)sin
Type: MC Var: 1
89) Which of these is the general solution of the second-order nonhomogeneous differential equation + 6 + 13y = 7cos(2t) + 
? Here, , , and all capital letters are arbitrary real constants.
A) y(t) = (cos(2t) + sin(2t)) + (At + B)cos(2t) + (Ct + D)sin(2t) + (E + Ft + G)
B) y(t) = (cos(2t) + sin(2t)) + Acos(2t) + Bsin(2t) + C
C) y(t) = (cos(2t) + sin(2t)) + (At + B)cos(2t) + (Ct + D)sin(2t) + E
D) y(t) = (cos(3t) + sin(3t)) + (At + B)cos(3t) + (Ct + D)sin(3t) + (E + Ft + G)
E) y(t) = (cos(3t) + sin(3t)) + (At + B)cos(3t) + (Ct + D)sin(3t) + E
Type: MC Var: 1
90) Which of the following functions Y(t) is a particular solution of the differential equation 
?
A) Y(t) = t + 1
B) Y(t) = 3t - 3
C) Y(t) = 3 - 3
D) Y(t) = - 1
Type: MC Var: 1
91) Which of the following functions Y(t) is a particular solution of the differential equation 
?
A) Y(t) = 9sin t - 9cos t
B) Y(t) = 
sin t - 
cos t
C) Y(t) = 
sin t - 
cos t
D) Y(t) = - sin t + cos t
Type: MC Var: 1
92) Consider this initial value problem:
+ - 2y = -2 + 2t + 14, y(0) = 0, (0) = 0
What is the general solution of the corresponding homogeneous differential equation?
Type: SA Var: 1
93) Consider this initial value problem:
+ - 2y = -2 + 2t + 14, y(0) = 0, (0) = 0
Find a particular solution of the given nonhomogeneous differential equation.
Type: SA Var: 1
94) Consider this initial value problem:
+ - 2y = -2 + 2t + 14, y(0) = 0, (0) = 0
What is the general solution of the nonhomogeneous differential equation?
Type: SA Var: 1
95) Consider this initial value problem:
+ - 2y = -2 + 2t + 14, y(0) = 0, (0) = 0
What is the solution of the initial value problem?
Type: SA Var: 1
96) Consider the nonhomogeneous differential equation + 2 + y = 3
.
Assume that 
= 
(t) + 
t is a particular solution of this differential equation. Use variation of parameters to determine (t) and (t).
(t) = ________ (t) = ________
Type: ES Var: 1
97) Consider the nonhomogeneous differential equation + 2 + y = 7.
What is the general solution of this differential equation?
Type: SA Var: 1
98) Use variation of parameters to find the general solution of the nonhomogeneous differential equation + - 6y = 7t.
A) y(t) = + + 
B) y(t) = + + 
(3t + 1)
C) y(t) = + + 
(6t + 1)
D) y(t) = + - (6t + 1)
Type: MC Var: 1
99) Use variation of parameters to find the general solution of the nonhomogeneous differential equation - 4 + 4y = sin(4t).
A) y(t) = ( + t) + 
(cos(4t) - sin(4t))
B) y(t) = ( + t) - sin(4t)
C) y(t) = ( + t) - sin(4t)
D) y(t) = ( + t) + (cos(4t) - sin(4t))
Type: MC Var: 1
100) Use variation of parameters to find the general solution of the nonhomogeneous Cauchy Euler differential equation - 4t + 6y = , t > 0.
A) y(t) = + -
B) y(t) = + -
C) y(t) = + + 
D) y(t) = + + 
E) y(t) = 
+ 
+
Type: MC Var: 1
101) Use variation of parameters to find the general solution of the nonhomogeneous differential equation - 2 + y = 
.
A) y(t) = + t - 
ln(1 + 4) + t
(2t)
B) y(t) = + t - tln(1 + 4) + (2t)
C) y(t) = + t - 8ln(1 + 4) + 2t(2t)
D) y(t) = + t - 8tln(1 + 4) + 2(2t)
Type: MC Var: 1
102) Use variation of parameters to find the general solution of the nonhomogeneous differential equation + 49y = sec(7t)
A) y(t) = cos(7t) + sin(7t) + 7sin(7t)ln|cos(7t)| + 7t cos(7t)
B) y(t) = cos(7t) + sin(7t) + sin(7t)ln|cos(7t)| + t cos(7t)
C) y(t) = cos(7t) + sin(7t) + 
cos(7t)ln|cos(7t)| + 
t sin(7t)
D) y(t) = cos(7t) + sin(7t) - cos(7t)ln|cos(7t)| + 7t sin(7t)
Type: MC Var: 1
103) Use variation of parameters to find the general solution of the nonhomogeneous differential equation + 25y = 
(5t)
A) y(t) = cos(5t) + sin(5t) + 
(5t) - 
(5t) + 
(5t)
B) y(t) = cos(5t) + sin(5t) + (5t) - (5t) + (5t)
C) y(t) = cos(5t) + sin(5t) + 
sin(5t)cos(5t)
+ 
(5t)cos(5t)
D) y(t) = cos(5t) + sin(5t) + sin(5t)cos(5t) + (5t)cos(5t)
Type: MC Var: 1
104) Suppose a 96-lb object stretches a spring 12 feet while in equilibrium and a dashpot provides a damping force of c libs for every foot per second of velocity. The equation of unforced motion of the object in such a spring-mass system is m + c + ky = 0, where m is the mass of the object, c is the damping constant, and k is the spring constant.
Determine the value of the constants m and k. (Use 1 slug = 32 lbs.)
m = ________ lb∙
/ foot
k = ________ lbs per foot
Type: ES Var: 1
105) Suppose a 96-lb object stretches a spring 4 feet while in equilibrium and a dashpot provides a damping force of c libs for every foot per second of velocity. The equation of unforced motion of the object in such a spring-mass system is m + c + ky = 0, where m is the mass of the object, c is the damping constant, and k is the spring constant.
For what values of the damping constant c is the motion critically damped? Round your answer to the nearest tenth.
c = ________ lb∙sec per foot
Type: SA Var: 1
106) Suppose a 64-lb object stretches a spring 2 feet while in equilibrium, and a dashpot provides a damping force of 3 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 m + c + ky = 0, 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 
feet below the horizontal with an upward velocity of - feet per second.
For what values of α and β is the function y(t) = (cos(βt) + sin(βt)) the general solution of the equation of motion for this spring-mass system? Provide exact values, not decimal approximations.
α = _______________, β = _______________
Type: ES Var: 1
107) 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 m + c + ky = 0, 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 feet below the horizontal with an upward velocity of - 3 feet per second.
For what values of the arbitrary constants and does a general solution of the form y(t) = 
satisfy the initial conditions? Provide the exact values, not decimal approximations.
= _______________, = _______________
Type: ES Var: 1
108) Suppose a 160-lb object stretches a spring 4 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 m + c + ky = 0, 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 feet below the horizontal with an upward velocity of - 3 feet per second.
What is the amplitude, R, (in feet) of the solution curve? Provide an exact value, not a decimal approximation.
R = ____________________ feet
Type: SA Var: 1
109) 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 m + c + ky = 0, 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 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, y(t), be expressed in the form 
. Provide the exact values, not decimal approximations.
A = ____________, B = ____________, C = ____________, tan D = ____________
Type: ES Var: 1
110) 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 m + c + ky = 0, 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 feet below the horizontal with an upward velocity of - feet per second.
What is the quasi-frequency? Provide the exact values, not a decimal approximation.
Type: SA Var: 1
111) Suppose a 160-lb object stretches a spring 3 feet while in equilibrium, and a dashpot provides a damping force of 3 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 m + c + ky = 0, 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 feet below the horizontal with an upward velocity of - 3 feet per second.
After how many seconds does the object pass through the equilibrium position for the first time? Round your answer to the nearest hundredth of a second.
Type: SA Var: 1
112) 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.
+ αu = 0, u(0) = β, 
(0) = γ
Type: ES Var: 1
113) Suppose a 6-lb object stretches a spring 1.75 feet while in equilibrium. If the object is displaced an additional 1 inches and is then set in motion with an initial upward velocity of -0.8 feet per second.
For what values of α and β is the function y(t) = (cos(βt) + sin(βt)) the general solution of the equation of motion for this spring-mass system? Round your answer to three decimal places.
α = ________, β = ________
Type: ES Var: 1
114) Suppose a 12-lb object stretches a spring 2 feet while in equilibrium. If the object is displaced an additional 2 inches and is then set in motion with an initial upward velocity of -1 feet per second.
For what values of the arbitrary constants and does the general solution 
satisfy the initial conditions? Provide the exact values, not decimal approximations.
= ________, = ________
Type: ES Var: 1
115) 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.
Type: SA Var: 1
116) Suppose a 6-lb object stretches a spring 1.75 feet while in equilibrium. If the object is displaced an additional 2 inches and is then set in motion with an initial upward velocity of -0.6 feet per second.
What is the period, T, in seconds? Round your answer to the nearest hundredth of a second.
Type: SA Var: 1
117) 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.
Type: SA Var: 1
118) Suppose a 10-lb object stretches a spring 2.25 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.
Suppose the phase angle δ is such that the solution curve can be expressed in the form 
. What is an expression for tan δ? Provide the exact value, not a decimal approximation.
tan δ = ________
Type: SA Var: 1
119) 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:
A + B + Cu = D(t), u(0) = E, (0) = F
Round all numerical values to the nearest hundredth, if needed.
Type: ES Var: 1
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Complete Test Bank | Differential Equations 12e
By William E. Boyce