Exam Prep Chapter.6 The Laplace Transform - Complete Test Bank | Differential Equations 12e by William E. Boyce. DOCX document preview.

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Exam Prep Chapter.6 The Laplace Transform

Elementary Differential Equations, 12e (Boyce)

Chapter 6 The Laplace Transform

1) Consider the following function:

f(t) = {((t - 7)) with superscript (2), 0 ≤ t < 8 2, 8 ≤ t ≤ 17 (2/17)t, t > 17

Which of these properties does f satisfy? Select all that apply.

A) f is piecewise continuous on [0, ∞).

B) f is of exponential order.

C) f is continuous on [0, ∞).

D) integral of ((e) with superscript (-st)f(t) dt) from (0) to (∞) diverges.

E) The Laplace transform of f exists.

Type: MC Var: 1

2) Consider the following function:

f(t) = {2.5, 0 ≤ t < 5 3.1(e) with superscript (-(t + 5)), t ≥ 5

Which of the following statements is true?

A) The Laplace transform of f does not exist because f is not of exponential order.

B) integral of ((e) with superscript (-st)f(t) dt) from (0) to (∞) converges.

C) f is continuous on [0, ∞)

D) f is piecewise continuous on [0, ∞), but the Laplace transform of f does not exist.

Type: MC Var: 1

3) Compute the Laplace transform of f:

f(t) = {4, 0 ≤ t < 5 0, t ≥ 5

A) (1 - (e) with superscript (-5s)/4), s > 0

B) ((e) with superscript (-5s) - 1/4), s > 0

C) ((e) with superscript (-5s) - 1/4s), s > 0

D) (4(1 - (e) with superscript (-5s))/s), s > 0

Type: MC Var: 1

4) Compute the Laplace transform of f:

f(t) = {12, 0 ≤ t ≤ 7 33 - 3t, t > 7

A) (12 + 12(e) with superscript (-7s)/s) + (33(e) with superscript (-7s)/s) + 3(e) with superscript (-7s)((1 + 7s/(s) with superscript (2)))

B) (12 - 12(e) with superscript (-7s)/s) + (33(e) with superscript (-7s)/s) - 3(e) with superscript (-7s)((1 + 7s/(s) with superscript (2)))

C) (12 - 12(e) with superscript (-7s)/(s) with superscript (2)) + (33(e) with superscript (-7s)/(s) with superscript (2)) - 3(e) with superscript (-7s)((1 + 7s/s))

D) (12 - 12(e) with superscript (-7s)/(s) with superscript (2)) - (33(e) with superscript (-7s)/(s) with superscript (2)) + 3(e) with superscript (-7s)((1 + 7s/s))

Type: MC Var: 1

5) The integral integral of ((((t) with superscript (2) + 1)) with superscript (-1)dt) from (0) to (∞) converges.

Type: TF Var: 1

6) Which of these statements is true?

A) integral of ((t) with superscript ( -4)(e) with superscript (4t)dt) from (0) to (∞)converges because ((e) with superscript (4t)/(t) with superscript (4)) ≤ (e) with superscript (4t), for all t ≥ 1 andintegral of ((e) with superscript (4t)dt) from (0) to (∞) converges.

B) integral of ((t) with superscript (2)(e) with superscript (-5t)dt) from (0) to (∞)diverges because (t) with superscript (2)(e) with superscript (-5t) ≥ (t) with superscript (2), for all t ≥ 1 andintegral of ((t) with superscript (2)dt) from (0) to (∞) converges.

C) integral of ((t) with superscript (4)(e) with superscript (-2t)dt) from (0) to (∞)converges because (t) with superscript (4)(e) with superscript (-2t) is of exponential order.

D) integral of ((t) with superscript ( -4)(e) with superscript (2t)dt) from (0) to (∞)converges because (t) with superscript ( -4)(e) with superscript (2t) is of exponential order.

Type: MC Var: 1

7) Compute the Laplace transform of f(t) = 4.2.

A) (4.2/s), s > 0

B) 4.2s, s > 0

C) (4.2/s), s ∈ ℝ

D) 4.2s, s ∈ ℝ

Type: MC Var: 1

8) Compute the Laplace transform of f(t) = 6t - 5(e) with superscript (3t).

A) (6/s) + (-5/s - 3), s > 3

B) (6/(s) with superscript (2)) + (-5/s + 3), s > -3

C) (6/(s) with superscript (2)) + (-5/s- 3), s > 3

D) (6/s) + (-5/s + 3), s > -3

Type: MC Var: 1

9) Compute the Laplace transform of f(t) = 3 sin(6t) - 4 cos(6t).

A) (18/(s) with superscript (2) - 36) + (-4s/(s) with superscript (2) - 36), s > 6

B) (3s/(s) with superscript (2) - 36) + (-4/(s) with superscript (2) - 36), s > 6

C) (18/(s) with superscript (2) + 36) + (-4s/(s) with superscript (2) + 36), s > 0

D) (3s/(s) with superscript (2) + 36) + (-4/(s) with superscript (2) + 36), s > 0

Type: MC Var: 1

10) Compute the Laplace transform of f(t) = t cos(10t).

A) ((s) with superscript (2) - 100/(((s) with superscript (2) + 100)) with superscript (2))

B) (-2s/(((s) with superscript (2) + 100)) with superscript (2))

C) ((s) with superscript (2) - 100/(s) with superscript (2) + 100)

D) (-20s/(s) with superscript (2) + 100)

E) (-20s/(((s) with superscript (2) + 100)) with superscript (2))

Type: MC Var: 1

11) Compute the Laplace transform of f(t) = sin(2πt)cos(2πt).

A) (2πs/(((s) with superscript (2) + 4(π) with superscript (2))) with superscript (2))

B) (2π(s) with superscript (2)/(((s) with superscript (2) + 4(π) with superscript (2))) with superscript (2))

C) (4π/(s) with superscript (2) + 16(π) with superscript (2))

D) (2π/(s) with superscript (2) + 16(π) with superscript (2))

Type: MC Var: 1

12) Compute the Laplace transform of f(t) = 3 + 10(t) with superscript (8).

A) (3/(s) with superscript (2)) + (10 ∙ 8!/(s) with superscript (9)), s > 0

B) (3/s) + (10 ∙ 8!/(s) with superscript (9)), s > 0

C) (3/s) + (10 ∙ 8/(s) with superscript (9)), s > 0

D) (3/(s) with superscript (2)) + (10 ∙ 9!/(s) with superscript (8)), s > 0

Type: MC Var: 1

13) Compute the inverse Laplace transform of F(s) = (square root of (17)/(s) with superscript (4)) + (9/(s) with superscript (2) + 5).

A) square root of (17)(t) with superscript (3) + (9/5)sin(5t)

B) (square root of (17)/3!)(t) with superscript (3) + (9/square root of (5))cos(square root of (5)t)

C) (square root of (17)/3!)(t) with superscript (3) + (9/square root of (5))sin(square root of (5)t)

D) (square root of (17)/4!)(t) with superscript (4) + (9/square root of (5))cos(square root of (5)t)

E) square root of (17)(t) with superscript (3) + (9/5)cos(5t)

Type: MC Var: 1

14) Compute the inverse Laplace transform of F(s) = (1/(s) with superscript (2)((s) with superscript (2) + 5)).

A) (1/5)(t) with superscript (2) - (1/25)cos(5t)

B) (1/5)t - (1/25)sin(5t)

C) (1/5)(t) with superscript (2) - (1/25)sin(5t)

D) (1/5)t - (1/5square root of (5))cos(square root of (5)t)

E) (1/5)t - (1/5square root of (5))sin(square root of (5)t)

Type: MC Var: 1

15) Compute the inverse Laplace transform of F(s) = (s + 1/(s) with superscript (2) + 81).

A) sin(81t) + (1/81)cos(9t)

B) sin(9t) + (1/9)cos(9t)

C) cos(9t) + (1/9)sin(9t)

D) cos(81t) + (1/81)sin(9t)

Type: MC Var: 1

16) Compute the inverse Laplace transform of F(s) = (1/(s + 2)(s + 3)).

A) (e) with superscript (3t) - (e) with superscript (2t)

B) (e) with superscript (2t) + (e) with superscript (3t)

C) (e) with superscript (2t) - (e) with superscript (3t)

D) (e) with superscript (-3t) + (e) with superscript (-2t)

E) (e) with superscript (-2t) + (e) with superscript (-3t)

F) (e) with superscript (-2t) - (e) with superscript (-3t)

Type: MC Var: 1

17) Find the Laplace transform of the solution x(t) of the following initial value problem:

x'' + 7x' + 3x = 8(t) with superscript (4), x(0) = 4, x'(0) = -1

A) (4(s) with superscript (6) + 27(s) with superscript (5) + 8 ∙ 4!/(s) with superscript (5)((s) with superscript (2) + 7s + 3))

B) (27(s) with superscript (5) - 4(s) with superscript (6) + 8 ∙ 4!/(s) with superscript (5)((s) with superscript (2) + 7s + 3))

C) (27(s) with superscript (5) - 4(s) with superscript (6) + 8 ∙ 4!/(s) with superscript (4)((s) with superscript (2) + 7s + 3))

D) (4(s) with superscript (6) - 27(s) with superscript (5) + 8 ∙ 4!/(s) with superscript (4)((s) with superscript (2) + 7s + 3))

Type: MC Var: 1

18) Find the Laplace transform of the solution x(t) of the following initial value problem:

x''' + 3x'' + 4x' + 4x = 0, x(0) = 3, x'(0) = -1, x''(0) = 3

A) (3(s) with superscript (2) - 8s - 12/(s) with superscript (3) + 3(s) with superscript (2) + 4s + 4)

B) (3(s) with superscript (2) - 8s - 12/(s) with superscript (3) - 3(s) with superscript (2) - 4s - 4)

C) (3(s) with superscript (2) + 8s + 12/(s) with superscript (3) + 3(s) with superscript (2) + 4s + 4)

D) (3(s) with superscript (2) + 8s + 12/(s) with superscript (3) - 3(s) with superscript (2) - 4s - 4)

Type: MC Var: 1

19) Consider the following initial value problem:

x' = -2 + 7(t) with superscript (8), x(0) = 5

(i) Find the Laplace transform solution x(t) of this initial value problem.

A. (5/s) + (-2/(s) with superscript (2)) + (7 ∙ 8!/(s) with superscript (10))

B. 5 + (-2/s) + (7 ∙ 8!/(s) with superscript (9))

C. (5/s) + (-2/(s) with superscript (2)) + (7/(s) with superscript (9))

D. 5 + (-2/s) + (7/(s) with superscript (9))

(ii) Find the inverse Laplace transform of the answer in part (i) to find the solution x(t) of the initial value problem.

(ii) 5 - 2t + (7/9)(t) with superscript (9)

Type: ES Var: 1

20) Find the Laplace transform of the solution x(t) of the following initial value problem:

x'' - 12x' + 32x = 0, x(0) = 1, x'(0) = 0

A) (1/s(s - 4)(s - 8))

B) (1/(s - 4)(s - 8))

C) (1s/(s - 4)(s - 8))

D) (1/s(s + 4)(s + 8))

E) (1s/(s + 4)(s + 8))

Type: MC Var: 1

21) Compute the Laplace transform of f(t) = t(e) with superscript (5t)sin(5t).

A) ((s) with superscript (2) - 20s + 75/((s - 5)) with superscript (2) + 25)

B) (5/((s - 5)) with superscript (2) + 25)

C) (-10(s - 5)/((((s - 5)) with superscript (2) + 25)) with superscript (2))

D) (10(s - 5)/((((s - 5)) with superscript (2) + 25)) with superscript (2))

Type: MC Var: 1

22) Compute the Laplace transform of f(t) = (t + 4)(U) with subscript (13)(t).

A) s(e) with superscript (-13t)((1/(s) with superscript (2)) + (4/s))

B) (e) with superscript (-13t)((1/(s) with superscript (2)) + (17/s))

C) (e) with superscript (-13t)((2/(s) with superscript (2)) - (17/s))

D) (e) with superscript (13t)((1/(s) with superscript (2)) + (17/s))

E) s(e) with superscript (13t)((1/(s) with superscript (2)) + (17/s))

Type: MC Var: 1

23) Compute the Laplace transform of f(t) = (t) with superscript (2)(U) with subscript (7)(t).

A) (e) with superscript (-7s)((2/(s) with superscript (3)) + (14/(s) with superscript (2)) + (49/s))

B) (e) with superscript (7s)((2/(s) with superscript (3)) + (14/(s) with superscript (2)) + (49/s))

C) (2/(s) with superscript (3)) + (14/(s) with superscript (2)) + (49/s)

D) (e) with superscript (7s)((2/(s) with superscript (3)) - (14/(s) with superscript (2)) - (49/s))

E) (e) with superscript (-7s)((2/(s) with superscript (3)) - (14/(s) with superscript (2)) - (49/s))

Type: MC Var: 1

24) Consider the function

f(t) = {t, 0 ≤ t ≤ 3 12, t > 3

Express f(t) using unit step functions.

A) t - (U) with subscript (3)(t)(t - 3)

B) t + (U) with subscript (12)(t)(t - 3)

C) t - (U) with subscript (3)(t)(t - 12)

D) t + (U) with subscript (3)(t)(t - 12)

Type: MC Var: 1

25) Consider the function

f(t) = {t, 0 ≤ t ≤ 6 13, t > 6

Compute the Laplace transform of f(t).

A) (1/(s) with superscript (2)) + (e) with superscript (-6s)((1/(s) with superscript (2)) + (7/s))

B) (1/(s) with superscript (2)) - (e) with superscript (-6s)((1/(s) with superscript (2)) - (7/s))

C) (1/(s) with superscript (2)) - (e) with superscript (6s)((1/(s) with superscript (2)) + (7/s))

D) (1/(s) with superscript (2)) + (e) with superscript (-6s)((1/(s) with superscript (2)) - (7/s))

Type: MC Var: 1

26) Consider the function

f(t) = {cos(6πt), 0 ≤ t ≤ 6 0, t > 6

Express f(t) using unit step functions.

A) cos(6πt)(U) with subscript (6)(t)

B) cos(6πt)(1 + (U) with subscript (6)(t))

C) cos(6πt)((U) with subscript (6)(t) - 1)

D) cos(6πt)(1 - (U) with subscript (6)(t))

Type: MC Var: 1

27) Consider the function

f(t) = {cos(2πt), 0 ≤ t ≤ 10 0, t > 10

Compute the Laplace transform of f(t).

A) (2π/4(π) with superscript (2) + (s) with superscript (2))(1 - (e) with superscript (10s))

B) (s/4(π) with superscript (2) + (s) with superscript (2))(1 - (e) with superscript (10s))

C) (2π/4(π) with superscript (2) + (s) with superscript (2))(1 - (e) with superscript (-10s))

D) (s/4(π) with superscript (2) + (s) with superscript (2))((e) with superscript (-10s) - 1)

E) (s/4(π) with superscript (2) + (s) with superscript (2))((e) with superscript (10s) - 1)

Type: MC Var: 1

28) Consider the function

f(t) = {0, 0 ≤ t < 8.5 t - 8.5, 8.5 ≤ t ≤ 17 0, t > 17

Express f(t) using unit step functions.

A) ((U) with subscript (8.5)(t) - (U) with subscript (17)(t))(t - 8.5)

B) ((U) with subscript (17)(t) - (U) with subscript (8.5)(t))(t - 8.5)

C) ((U) with subscript (8.5)(t) + (U) with subscript (17)(t))(t - 8.5)

D) - ((U) with subscript (8.5)(t) + (U) with subscript (17)(t))(t - 8.5)

Type: MC Var: 1

29) Consider the function

f(t) = {0, 0 ≤ t < 8.5 t - 8.5, 8.5 ≤ t ≤ 17 0, t > 17

Compute the Laplace transform of f(t).

A) ((e) with superscript (-8.5s)/(s) with superscript (2))((e) with superscript (-8.5s) - 1) + (8.5(e) with superscript (-17s)/s)

B) ((e) with superscript (-8.5s)/s)((e) with superscript (-8.5s) - 1) - (8.5(e) with superscript (-17s)/(s) with superscript (2))

C) ((e) with superscript (-8.5s)/s)(1 - (e) with superscript (-8.5s)) + (8.5(e) with superscript (-17s)/(s) with superscript (2))

D) ((e) with superscript (-8.5s)/(s) with superscript (2))(1 - (e) with superscript (-8.5s)) - (8.5(e) with superscript (-17s)/s)

Type: MC Var: 1

30) Compute the inverse Laplace transform of F(s) = (s + 2/(s) with superscript (2) - 4s + 15).

A) (e) with superscript (-2t)(cos(square root of (11)t) + (4/square root of (11))sin(square root of (11)t))

B) (e) with superscript (2t)(cos(square root of (11)t) + (4/square root of (11))sin(square root of (11)t))

C) (e) with superscript (2t)((4/square root of (11))cos(square root of (11)t) + sin(square root of (11)t))

D) (e) with superscript (-2t)((4/square root of (11))cos(square root of (11)t) + sin(square root of (11)t))

Type: MC Var: 1

31) Consider the function

f(t) = {3, 0 ≤ t < 5 0, 5 ≤ t ≤ 6 and f(t + 6) = f(t), for all t ≥ 0.

Compute the Laplace transform of f(t).

A) (3(1 - (e) with superscript (-5s))/s(1 - (e) with superscript (-6s)))

B) (3((e) with superscript (-5s) - 1)/s(1 - (e) with superscript (-6s)))

C) (3(1 - (e) with superscript (-5s))/s(1 - (e) with superscript (6s)))

D) (3((e) with superscript (-5s) - 1)/s(1 - (e) with superscript (6s)))

Type: MC Var: 1

32) Compute the inverse Laplace transform of F(s) = ((e) with superscript (-3s)(s - 3)/((s - 3)) with superscript (2) + 16).

A) (U) with subscript (-3)(t)(e) with superscript (3t)cos(4t)

B) (U) with subscript (3)(t)(e) with superscript (3t)cos(4t)

C) (U) with subscript (3)(t)(e) with superscript (-3t)cos(4t)

D) (U) with subscript (3)(t)(e) with superscript (3t)sin(4t)

E) (U) with subscript (3)(t)(e) with superscript (-3t)sin(4t)

F) (U) with subscript (-3)(t)(e) with superscript (-3t)sin(4t)

Type: MC Var: 1

33) Compute the Laplace transform of f(t) = (t) with superscript ( 8)(e) with superscript (-6t).

A) (8!/((s + 6)) with superscript (9))

B) (8!/((s - 6)) with superscript (9))

C) (8!/(s) with superscript (9)(s + 6))

D) (8!/(s) with superscript (9)(s - 6))

Type: MC Var: 1

34) Compute the Laplace transform of f(t) = (e) with superscript (4t)cos(4t).

A) (s + 4/((s + 4)) with superscript (2) + 16)

B) (1/((s + 4)) with superscript (2) + 16)

C) (1/((s - 4)) with superscript (2) + 16)

D) (s - 4/((s - 4)) with superscript (2) + 16)

Type: MC Var: 1

35) Compute the inverse Laplace transform of F(s) = (1/((s + 4)) with superscript (4)).

A) (e) with superscript (-4t)(t) with superscript ( 4)

B) (e) with superscript (-4t)(t) with superscript ( 3)

C) ((e) with superscript (-4t)(t) with superscript ( 3)/3!)

D) ((e) with superscript (4t)(t) with superscript ( 4)/4!)

E) ((e) with superscript (4t)((t + 4)) with superscript (3)/3!)

Type: MC Var: 1

36) Consider the function

f(t) = {8, 0 ≤ t ≤ 6π 0, t > 6π

Express f(t) using unit step functions.

A) 8 - 8(U) with subscript (6π)(t)

B) 8 + 8(U) with subscript (6π)(t)

C) 8 - 8(U) with subscript (-6π)(t)

D) 8 + 8(U) with subscript (-6π)(t)

Type: MC Var: 1

37) Compute the inverse Laplace transform of F(s) = (3s + 7/((s + 5)) with superscript (4)).

A) (e) with superscript (-5t)((3/2!)(t) with superscript ( 2) - (8/3!)(t) with superscript ( 3))

B) (e) with superscript (5t)((3/2!)(t) with superscript ( 2) - (8/3!)(t) with superscript ( 3))

C) (e) with superscript (-5t)((3/2!)(t) with superscript ( 2) - (2/3!)(t) with superscript ( 3))

D) (e) with superscript (5t)((3/2!)(t) with superscript ( 2) - (2/3!)(t) with superscript ( 3))

Type: MC Var: 1

38) Find the Laplace transform of the solution x(t) of the following initial value problem:

x'' + 7x' + 5x = f(t), x(0) = -5, x'(0) = -5

where

f(t) = {cos t, 0 ≤ t < 8π t - 8π, t ≥ 8π

A) (1/(s) with superscript (2) + 7s + 5)(5s + 40 + (((e) with superscript (-8πs) - 1)/(s) with superscript (2) + 1) + ((e) with superscript (-gπs)/(s) with superscript (2)))

B) (1/(s) with superscript (2) + 7s + 5)(-5s - 40 + (((e) with superscript (-8πs) - 1)/(s) with superscript (2) + 1) + ((e) with superscript (-gπs)/(s) with superscript (2)))

C) (1/(s) with superscript (2) + 7s + 5)(5s + 40 + ((1 - (e) with superscript (-8πs))s/(s) with superscript (2) + 1) + ((e) with superscript (-gπs)/(s) with superscript (2)))

D) (1/(s) with superscript (2) + 7s + 5)(-5s - 40 + ((1 - (e) with superscript (-8πs))s/(s) with superscript (2) + 1) + ((e) with superscript (-gπs)/(s) with superscript (2)))

Type: MC Var: 1

39) Find the Laplace transform of the solution x(t) of the following initial value problem

x'' - 3x' - 3x = (t) with superscript (2) + (U) with subscript (6)(t)(-2t - (t) with superscript (2)), x(0) = -3, x'(0) = -4

A) (1/(s) with superscript (2) - 3s - 3)(-3s + 5 + (2/(s) with superscript (3)) + (e) with superscript (6s)- (2/(s) with superscript (3)) + (-14/(s) with superscript (2)) + (-48/s))

B) (1/(s) with superscript (2) - 3s - 3)(-3s + 5 + (2/(s) with superscript (3)) + (e) with superscript (-6s)- (2/(s) with superscript (3)) + (-14/(s) with superscript (2)) + (-48/s))

C) (1/(s) with superscript (2) - 3s - 3)(3s - 5 + (2/(s) with superscript (3)) + (e) with superscript (-6s)- (2/(s) with superscript (3)) + (-14/(s) with superscript (2)) + (-48/s))

D) (1/(s) with superscript (2) - 3s - 3)(3s - 5 + (2/(s) with superscript (3)) + (e) with superscript (6s)- (2/(s) with superscript (3)) + (-14/(s) with superscript (2)) + (-48/s))

E) (1/(s) with superscript (2) - 3s - 3)(-3s + 5 + (2/(s) with superscript (3)) + (e) with superscript (-6s)(2/(s) with superscript (3)) + (14/(s) with superscript (2)) + (48/s))

F) (1/(s) with superscript (2) - 3s - 3)(3s - 5 + (2/(s) with superscript (3)) + (e) with superscript (-6s)(2/(s) with superscript (3)) + (14/(s) with superscript (2)) + (48/s))

Type: MC Var: 1

40) Find the Laplace transform of the solution x(t) of the following initial value problem

x' = 5x + (U) with subscript (6)(t), x(0) = -3

A) (-3/s - 5) + ((e) with superscript (-6s)/s(s - 5))

B) (-3/s + 5) + ((e) with superscript (-6s)/s(s + 5))

C) - (-3/s - 5) + ((e) with superscript (6s)/s(s - 5))

D) (-3/s + 5) + ((e) with superscript (6s)/s(s + 5))

Type: MC Var: 1

41) Find the Laplace transform of the solution x(t) of the following initial value problem

x'' + 9x = (U) with subscript (3)(t)((t - 3)) with superscript (10)

A) ((e) with superscript (-9s) ∙ 9!/(s) with superscript (10)((s) with superscript (2) + 9))

B) ((e) with superscript (9s) ∙ 9!/(s) with superscript (10)((s) with superscript (2) + 9))

C) ((e) with superscript (9s) ∙ 10!/(s) with superscript (11)((s) with superscript (2) + 9))

D) ((e) with superscript (-9s) ∙ 10!/(s) with superscript (11)((s) with superscript (2) + 9))

Type: MC Var: 1

42) You are given a spring-mass system with a mass of 1 slug, a damping constant 8 lb-sec/foot, and a spring constant of 16 lbs/foot. Suppose the mass is released from rest 1.5 feet below equilibrium, and after 5π seconds the system is given a sharp blow downward which imparts a unit impulse.

(i) Write down a second-order initial value problem whose solution x(t) is the equation of motion for this system.

(ii) Find the Laplace transform X(s) of the solution x(t) of the initial value problem you formulated in part (i).

(iii) Compute the inverse Laplace transform of your function in part (ii).

(ii) X(s) = (1.5s + 12 + (e) with superscript (-5πs)/((s + 4)) with superscript (2))

(iii) x(t) = 1.5(e) with superscript (-4t) + 6t(e) with superscript (-4t) + (t - 5π)(e) with superscript (-4(t - 5π))(U) with subscript (5π)(t)

Type: ES Var: 1

43) integral of (δ(t - 1)(-3(t) with superscript ( 3) - 2(t) with superscript (2) + 6) dt) from (-∞) to (∞) = ________. Here, δ stands for the Dirac delta function.

Type: SA Var: 1

44) integral of (δ(t - 6)sin4πt + (π/2) dt) from (-∞) to (∞) = ________. Here, δ stands for the Dirac delta function.

Type: SA Var: 1

45) Compute the Laplace transform of f(t) = δ(t + 3), where δ stands for the Dirac delta function.

Type: SA Var: 1

46) Find the Laplace transform of the solution of x(t) of the following initial value problem

x'' + 4x' + 4x = -4δ(t - 5), x(0) = 5, x'(0) = 0

A) (5s + 20 - 4(e) with superscript (5s)/((s + 2)) with superscript (2))

B) (5s + 20 - 4(e) with superscript (-5s)/((s + 2)) with superscript (2))

C) (25 - 4(e) with superscript (5s)/((s + 2)) with superscript (2))

D) (25 - 4(e) with superscript (-5s)/((s + 2)) with superscript (2))

Type: MC Var: 1

47) Consider the following initial value problem

x'' + 4x' + 4x = δ(t - 5) - δ(t - 6), x(0) = 3, x'(0) = 5

(i) Find the Laplace transform X(s) of the solution x(t) of this initial value problem.

(ii) Compute the inverse Laplace transform of your function in part (i).

(ii) x(t) = 3(e) with superscript (-2t) + 11t(e) with superscript (-2t) + (U) with subscript (5)(t)(t - 5)(e) with superscript (-(t - 5)) + (U) with subscript (6)(t)(t - 6)(e) with superscript (-(t - 6))

Type: ES Var: 1

48) Compute (e) with superscript (2t) * (e) with superscript (-5t).

A) (1/7)((e) with superscript (5t) - (e) with superscript (-2t))

B) (1/7)((e) with superscript (2t) - (e) with superscript (-5t))

C) (1/7)((e) with superscript (-2t) - (e) with superscript (5t))

D) (1/7)((e) with superscript (-5t) - (e) with superscript (2t))

Type: MC Var: 1

49) Which of the following are properties of the convolution integral, for all continuous functions f, g, and h? Select all that apply.

A) f * (g * h) = (f * g) * h

B) f * f ≥ 0

C) f * 1 = f

D) f * g = g * f

E) f * (g + h) = f * g + f * h

Type: MC Var: 1

50) Compute the Laplace transform of f(t) = integral of (s(e) with superscript (6s)ds) from (0) to (t).

A) (1/s) - (1/s - 6)

B) (1/s) + (1/((s + 6)) with superscript (2))

C) (1/s) + (1/((s - 6)) with superscript (2))

D) (1/s((s - 6)) with superscript (2))

E) (1/s((s + 6)) with superscript (2))

F) - (1/s(s - 6))

Type: MC Var: 1

51) Consider the following initial value problem describing the motion of a harmonic oscillator in the absence of friction, but subject to an external force.

x'' + 6x = f(t), x(0) = 1.6, x'(0) = 1.4

Find the Laplace transform X(s) of the solution x(t) of this initial value problem. Here, F(s) stands for the Laplace transform of f(t).

A) (1.6s + 1.4 + F(s)/(s) with superscript (2) + 6)

B) (-1.6s - 1.4 + F(s)/(s) with superscript (2) + 6)

C) (1.6s - 1.4 + F(s)/(s) with superscript (2) + 6)

D) (1.4 - 1.6s + F(s)/(s) with superscript (2) + 6)

Type: MC Var: 1

52) Consider the following initial value problem describing the motion of a harmonic oscillator in the absence of friction, but subject to an external force.

x'' + 6x = f(t), x(0) = -1.4, x'(0) = -1.8

Find the equation of motion x(t). (Hint: You will need to use a convolution integral.)

Type: SA Var: 1

53) Find the function f(t) that satisfies the integral equation

f(t) = -5t + integral of (f(s)sin(t - s)ds) from (0) to (t)

Type: SA Var: 1

54) Compute (t) with superscript ( 3) * t.

Type: SA Var: 1

55) Use the convolution theorem to compute the inverse Laplace transform of F(s) = (1/(s) with superscript (5)(s - 8))

A) ((t) with superscript ( 4)/4!) * (e) with superscript (8t)

B) ((t) with superscript ( 5)/5!) * (e) with superscript (8t)

C) ((t) with superscript ( 4)/4!) * (e) with superscript (-8t)

D) ((t) with superscript ( 5)/5!) * (e) with superscript (-8t)

E) (t) with superscript ( 5) * (e) with superscript (8t)

F) (t) with superscript ( 5) * (e) with superscript (-8t)

Type: MC Var: 1

56) Find the function f(t) that satisfies the integral equation

f(t) + integral of (f(s) ds) from (0) to (t) = 1

Type: SA Var: 1

57) Use the convolution theorem to compute the inverse Laplace transform of F(s) = (28s/((s) with superscript (2) + 16)((s) with superscript (2) + 49)). Select all that apply.

A) 28 sin(4t) * cos(7t)

B) 7 sin(4t) * cos(7t)

C) 4 sin(4t) * cos(7t)

D) 4 sin(7t) * cos(4t)

E) 7 sin(7t) * cos(4t)

Type: MC Var: 1

© (2022) John Wiley & Sons, Inc. All rights reserved. Instructors who are authorized users of this course are permitted to download these materials and use them in connection with the course. Except as permitted herein or by law, no part of these materials should be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise.

Document Information

Document Type:
DOCX
Chapter Number:
6
Created Date:
Aug 21, 2025
Chapter Name:
Chapter 6 The Laplace Transform
Author:
William E. Boyce

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