Deck 6: The Laplace Transform

Question 1

Consider the following function:
F(t) =
Which of these properties does f satisfy? Select all that apply.

A) f is piecewise continuous on [0, $\infty$ ).
B) f is of exponential order.
C) f is continuous on [0, $\infty$ ).
D) diverges.
E) The Laplace transform of f exists.

Answer 1

f is piecewise continuous on [0, $\infty$ ).
f is of exponential order.
The Laplace transform of f exists.

Question 2

Consider the following function:
F(t) =
Which of the following statements is true?

A) The Laplace transform of f does not exist because f is not of exponential order.
B) converges.
C) f is continuous on [0, $\infty$ )
D) f is piecewise continuous on [0, $\infty$ ), but the Laplace transform of f does not exist.

Answer 2

converges.

Question 3

Compute the Laplace transform of f:
F(t) =

A) , s > 0
B) , s > 0
C) , s > 0
D) , s > 0

Answer 3

, s > 0

Question 4

Compute the Laplace transform of f:
F(t) =

A) $\frac{12+12 e^{-7 s}}{s}+\frac{33 e^{-7 s}}{s}+3 e^{-7 s}\left(\frac{1+7 s}{s^{2}}\right)$
B) $\frac{12-12 e^{-7 s}}{s}+\frac{33 e^{-7 s}}{s}-3 e^{-7 s}\left(\frac{1+7 s}{s^{2}}\right)$
C) $\frac{12-12 e^{-7 s}}{s^{2}}+\frac{33 e^{-7 s}}{s^{2}}-3 e^{-7 s}\left(\frac{1+7 s}{s}\right)$
D) $\frac{12-12 e^{-7 s}}{s^{2}}-\frac{33 e^{-7 s}}{s^{2}}+3 e^{-7 s}\left(\frac{1+7 s}{s}\right)$

Question 5

The integral converges.

Question 6

Which of these statements is true?

A) $\int_{0}^{\infty} t^{-4} e^{4 t} d t$ converges because $\frac{e^{4 t}}{t^{4}} \leq e^{4 t}$ , for all $t \geq 1$ and $\int_{0}^{\infty} e^{4 t} d t$ converges.
B) $\int_{0}^{\infty} t^{2} e^{-5 t} d t$ diverges because $t^{2} e^{-5 t} \geq t^{2}$ , for all $t \geq 1$ and $\int_{0}^{\infty} t^{2} d t$ converges.
C) $\int_{0}^{\infty} A^{4} e^{-2 t} d t$ converges because $t^{t} e^{-2 t}$ is of exponential order.
D) $\int_{0}^{\infty} t^{-4} e^{2 t} d t$ converges because $t^{-4} e^{2 t}$ is of exponential order.

Question 7

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

A) $\frac{18}{s^{2}-36}+\frac{-4 s}{s^{2}-36}, s>6$
B) $\frac{3 s}{s^{2}-36}+\frac{-4}{s^{2}-36}, s>6$
C) $\frac{18}{s^{2}+36}+\frac{-4 s}{s^{2}+36}, s>0$
D) $\frac{3 s}{s^{2}+36}+\frac{-4}{s^{2}+36}, s>0$

Question 8

Compute the Laplace transform of f(t) = 6t - 5 .

A) + , s > 3
B) + , s > -3
C) + , s > 3
D) + , s > -3

Question 9

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

A) $\frac{18}{s^{2}-36}+\frac{-4 s}{s^{2}-36}, s>6$
B) $\frac{3 s}{s^{2}-36}+\frac{-4}{s^{2}-36}, s>6$
C) $\frac{18}{s^{2}+36}+\frac{-4 s}{s^{2}+36}, s>0$
D) $\frac{3 s}{s^{2}+36}+\frac{-4}{s^{2}+36}, s>0$

Question 10

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

A)
B)
C)
D)
E)

Question 11

Compute the Laplace transform of f(t) = sin(2 $\pi$ t)cos(2 $\pi$ t).

A)
B)
C)
D)

Question 12

Compute the Laplace transform of f(t) = 3 + 10 .

A) $\frac{3}{s^{2}}+\frac{10 \cdot 8 !}{s^{9}}, s>0$
B) $\frac{3}{s}+\frac{10 \cdot 8 !}{s^{9}}, s>0$
C) $\frac{3}{s}+\frac{10 \cdot 8}{s^{9}}, s>0$
D) $\frac{3}{s^{2}}+\frac{10 \cdot 9 !}{s^{8}}, s>0$

Question 13

Compute the inverse Laplace transform of F(s) = + .

A) $\sqrt{17} t^{3}+\frac{9}{5} \sin (5 t)$
B) $\frac{\sqrt{17}}{3 !} t^{3}+\frac{9}{\sqrt{5}} \cos (\sqrt{5} t)$
C) $\frac{\sqrt{17}}{3 !} t^{3}+\frac{9}{\sqrt{5}} \sin (\sqrt{5} t)$
D) $\frac{\sqrt{17}}{4 !} t+\frac{9}{\sqrt{5}} \cos (\sqrt{5})$
E) $\sqrt{17} t^{3}+\frac{9}{5} \cos (5 t)$

Question 14

Compute the inverse Laplace transform of

A) $\frac{1}{5} t^{2}-\frac{1}{25} \cos (5 t)$
B) $\frac{1}{5} t-\frac{1}{25} \sin (5 t)$
C) $\frac{1}{5} t^{2}-\frac{1}{25} \sin (5 t)$
D) $\frac{1}{5} t-\frac{1}{5 \sqrt{5}} \cos (\sqrt{5} t)$
E) $\frac{1}{5} t-\frac{1}{5 \sqrt{5}} \sin (\sqrt{5} t)$

Question 15

Compute the inverse Laplace transform of F(s) = .

A) $\sin (81 t)+\frac{1}{81} \cos (9 t)$
B) $\sin (9 t)+\frac{1}{9} \cos (9 t)$
C) $\cos (9 t)+\frac{1}{9} \sin (9 t)$
D) $\cos (81 t)+\frac{1}{81} \sin (9 t)$

Question 16

Compute the inverse Laplace transform of

A) $e^{3 t}-e^{2 t}$
B) $e^{2 t}+e^{3 t}$
C) $e^{2 t}-e^{3 t}$
D) $e^{-3 t}+e^{-2 t}$
E) $e^{-2 t}+e^{-3 t}$
F) $e^{-2 t}-e^{-3 t}$

Question 17

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

A) $\frac{4 s^{6}+27 s^{5}+8 \cdot 4 !}{s^{5}\left(s^{2}+7 s+3\right)}$
B) $\frac{27 s^{5}-4 s^{6}+8 \cdot 4 !}{s^{5}\left(s^{2}+7 s+3\right)}$
C) $\frac{27 s^{5}-4 s^{6}+8 \cdot 4 !}{s^{4}\left(s^{2}+7 s+3\right)}$
D) $\frac{4 s^{6}-27 s^{5}+8 \cdot 4 !}{s^{4}\left(s^{2}+7 s+3\right)}$

Question 18

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

A) $\frac{3 s^{2}-8 s-12}{s^{3}+3 s^{2}+4 s+4}$
B) $\frac{3 s^{2}-8 s-12}{s^{3}-3 s^{2}-4 s-4}$
C) $\frac{3 s^{2}+8 s+12}{s^{3}+3 s^{2}+4 s+4}$
D) $\frac{3 s^{2}+8 s+12}{s^{3}-3 s^{2}-4 s-4}$

Question 19

Consider the following initial value problem:

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

Question 20

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

A) $\frac{1}{s(s-4)(s-8)}$
B) $\frac{1}{(s-4)(s-8)}$
C) $\frac{1 s}{(s-4)(s-8)}$
D) $\frac{1}{s(s+4)(s+8)}$
E) $\frac{1 s}{(s+4)(s+8)}$

Question 21

Compute the Laplace transform of f(t) = sin(5t).

A) $\frac{s^{2}-20 s+75}{(s-5)^{2}+25}$
B) $\frac{5}{(s-5)^{2}+25}$
C) $\frac{-10(s-5)}{\left((s-5)^{2}+25\right)^{2}}$
D) $\frac{10(s-5)}{\left((s-5)^{2}+25\right)^{2}}$

Question 22

Compute the Laplace transform of

A) $s e^{-13 t}\left(\frac{1}{s^{2}}+\frac{4}{s}\right)$
B) $e^{-13 t}\left(\frac{1}{s^{2}}+\frac{17}{s}\right)$
C) $e^{-13 t}\left(\frac{2}{s^{2}}-\frac{17}{s}\right)$
D) $e^{13 t}\left(\frac{1}{s^{2}}+\frac{17}{s}\right)$
E) $s e^{13 t}\left(\frac{1}{s^{2}}+\frac{17}{s}\right)$

Question 23

Compute the Laplace transform of

A) $e^{-7}\left(\frac{2}{s^{3}}+\frac{14}{s^{2}}+\frac{49}{s}\right)$
B) $e^{7 s}\left(\frac{2}{s^{3}}+\frac{14}{s^{2}}+\frac{49}{s}\right)$
C) $\frac{2}{s^{3}}+\frac{14}{s^{2}}+\frac{49}{s}$
D) $e^{75}\left(\frac{2}{s^{3}}-\frac{14}{s^{2}}-\frac{49}{s}\right)$
E) $e^{-7}\left(\frac{2}{s^{3}}-\frac{14}{s^{2}}-\frac{49}{s}\right)$

Question 24

Consider the function

Express f(t) using unit step functions.

A) $t-U_{3}(t)(t-3)$
B) $t+U_{12}(t)(t-3)$
C) $t-U_{3}(t)(t-12)$
D) $t+U_{3}(t)(t-12)$

Question 25

Consider the function

Compute the Laplace transform of f(t).

A) $\frac{1}{s^{2}}+e^{-6 s}\left(\frac{1}{s^{2}}+\frac{7}{s}\right)$
B) $\frac{1}{s^{2}}-e^{-6 s}\left(\frac{1}{s^{2}}-\frac{7}{s}\right)$
C) $\frac{1}{s^{2}}-e^{6 s}\left(\frac{1}{s^{2}}+\frac{7}{s}\right)$
D) $\frac{1}{s^{2}}+e^{-6 s}\left(\frac{1}{s^{2}}-\frac{7}{s}\right)$

Question 26

Consider the function

Express f(t) using unit step functions.

A) $\cos (6 \pi t) U_{6}(t)$
B) $\cos (6 \pi t)\left(1+U_{6}(t)\right)$
C) $\cos (6 \pi t)\left(U_{6}(t)-1\right)$
D) $\cos (6 \pi t)\left(1-U_{6}(t)\right)$

Question 27

Consider the function

Compute the Laplace transform of f(t).

A) $\frac{2 \pi}{4 \pi^{2}+s^{2}}\left(1-e^{10 s}\right)$
B) $\frac{s}{4 \pi^{2}+s^{2}}\left(1-e^{10 s}\right)$
C) $\frac{2 \pi}{4 \pi^{2}+s^{2}}\left(1-e^{-10 s}\right)$
D) $\frac{s}{4 \pi^{2}+s^{2}}\left(e^{-10 s}-1\right)$
E) $\frac{s}{4 \pi^{2}+s^{2}}\left(e^{10 s}-1\right)$

Question 28

Consider the function

Express f(t) using unit step functions.

A) $\left(U_{8.5}(t)-U_{17}(t)\right)(t-8.5)$
B) $\left(U_{17}(t)-U_{8.5}(t)\right)(t-8.5)$
C) $\left(U_{8.5}(t)+U_{17}(t)\right)(t-8.5)$
D) $-\left(U_{85}(t)+U_{17}(t)\right)(t-8.5)$

Question 29

Consider the function

Compute the Laplace transform of f(t).

A) $\frac{e^{-8.5 s}}{s^{2}}\left(e^{-8.5 s}-1\right)+\frac{8.5 e^{-17 s}}{s}$
B) $\frac{e^{-8.5 s}}{s}\left(e^{-8.5 s}-1\right)-\frac{8.5 e^{-17 s}}{s^{2}}$
C) $\frac{e^{-8.5 s}}{s}\left(1-e^{-8.5 s}\right)+\frac{8.5 e^{-17 s}}{s^{2}}$
D) $\frac{e^{-8.5 s}}{s^{2}}\left(1-e^{-8.5 s}\right)-\frac{8.5 e^{-17 s}}{s}$

Question 30

Compute the inverse Laplace transform of F(s) = .

A) $e^{-2 t}\left[\cos (\sqrt{11} t)+\frac{4}{\sqrt{11}} \sin (\sqrt{11} t)\right]$
B) $e^{2 t}\left(\cos (\sqrt{11} t)+\frac{4}{\sqrt{11}} \sin (\sqrt{11} t)\right)$
C) $e^{2 t}\left(\frac{4}{\sqrt{11}} \cos (\sqrt{11} t)+\sin (\sqrt{11} t)\right)$
D) $e^{-2 t}\left(\frac{4}{\sqrt{11}} \cos (\sqrt{11} t)+\sin (\sqrt{11} t)\right)$

Question 31

Consider the function

Compute the Laplace transform of f(t).

A) $\frac{3\left(1-e^{-5 s}\right)}{s\left(1-e^{-6 s}\right)}$
B) $\frac{3\left(e^{-5 s}-1\right)}{s\left(1-e^{-6 s}\right)}$
C) $\frac{3\left(1-e^{-5 s}\right)}{s\left(1-e^{6 s}\right)}$
D) $\frac{3\left(e^{-5 s}-1\right)}{s\left(1-e^{65}\right)}$

Question 32

Compute the inverse Laplace transform of

A) $U_{-3}(t) e^{3 t} \cos (4 t)$
B) $U_{3}(t) e^{3 t} \cos (4 t)$
C) $U_{3}(t) e^{-3 t} \cos (4 t)$
D) $U_{3}(t) e^{3 t} \sin (4 t)$
E) $U_{3}(t) e^{-3 t} \sin (4 t)$
F) $U_{-3}(t) e^{-3 t} \sin (4 t)$

Question 33

Compute the Laplace transform of

A) $\frac{8 !}{(s+6)^{9}}$
B) $\frac{8 !}{(s-6)^{9}}$
C) $\frac{8 !}{s^{9}(s+6)}$
D) $\frac{8 !}{s^{9}(s-6)}$

Question 34

Compute the Laplace transform of

A) $\frac{s+4}{(s+4)^{2}+16}$
B) $\frac{1}{(s+4)^{2}+16}$
C) $\frac{1}{(s-4)^{2}+16}$
D) $\frac{s-4}{(s-4)^{2}+16}$

Question 35

Compute the inverse Laplace transform of .

A) $e^{-4 t} t^{4}$
B) $e^{-4 t} t^{3}$
C) $\frac{e^{-4 t} t^{3}}{3 !}$
D) $\frac{e^{4 t} t^{4}}{4 !}$
E) $\frac{e^{4 t}(t+4)^{3}}{3 !}$

Question 36

Consider the function

Express f(t) using unit step functions.

A) $e^{-4 t} t^{4}$
B) $e^{-4 t} t^{3}$
C) $\frac{e^{-4 t} t^{3}}{3 !}$
D) $\frac{e^{4 t} t^{4}}{4 !}$
E) $\frac{e^{4 t}(t+4)^{3}}{3 !}$

Question 37

Compute the inverse Laplace transform of

A) $e^{-5 t}\left(\frac{3}{2 !} t^{2}-\frac{8}{3 !} t^{3}\right)$
B) $e^{5 t}\left(\frac{3}{2 !} t^{2}-\frac{8}{3 !} t^{3}\right)$
C) $e^{-5 t}\left(\frac{3}{2 !} t^{2}-\frac{2}{3 !} t^{3}\right)$
D) $e^{5 t}\left(\frac{3}{2 !} t^{2}-\frac{2}{3 !} t^{3}\right)$

Question 38

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

Wher

A) $\frac{1}{s^{2}+7 s+5}\left(5 s+40+\frac{\left(e^{-8 \pi s}-1\right)}{s^{2}+1}+\frac{e^{-\mathrm{g} \pi s}}{s^{2}}\right)$
B) $\frac{1}{s^{2}+7 s+5}\left(-5 s-40+\frac{\left(e^{-8 \pi s}-1\right)}{s^{2}+1}+\frac{e^{-g \pi s}}{s^{2}}\right)$
C) $\frac{1}{s^{2}+7 s+5}\left(5 s+40+\frac{\left(1-e^{-8 \pi s}\right) s}{s^{2}+1}+\frac{e^{-\mathrm{g} \pi s}}{s^{2}}\right)$
D) $\frac{1}{s^{2}+7 s+5}\left(-5 s-40+\frac{\left(1-e^{-8 \pi s}\right) s}{s^{2}+1}+\frac{e^{-g \pi s}}{s^{2}}\right)$

Question 39

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

A) $\frac{1}{s^{2}-3 s-3}\left(-3 s+5+\frac{2}{s^{3}}+e^{6 s}-\frac{2}{s^{3}}+\frac{-14}{s^{2}}+\frac{-48}{s}\right)$
B) $\frac{1}{s^{2}-3 s-3}\left(-3 s+5+\frac{2}{s^{3}}+e^{-6 s}-\frac{2}{s^{3}}+\frac{-14}{s^{2}}+\frac{-48}{s}\right)$
C) $\frac{1}{s^{2}-3 s-3}\left(3 s-5+\frac{2}{s^{3}}+e^{-6 s}-\frac{2}{s^{3}}+\frac{-14}{s^{2}}+\frac{-48}{s}\right)$
D) $\frac{1}{s^{2}-3 s-3}\left(3 s-5+\frac{2}{s^{3}}+e^{6 s}-\frac{2}{s^{3}}+\frac{-14}{s^{2}}+\frac{-48}{s}\right)$
E) $\frac{1}{s^{2}-3 s-3}\left(-3 s+5+\frac{2}{s^{3}}+e^{-6 s} \frac{2}{s^{3}}+\frac{14}{s^{2}}+\frac{48}{s}\right)$
F) $\frac{1}{s^{2}-3 s-3}\left(3 s-5+\frac{2}{s^{3}}+e^{-6 s^{2}} s^{3}+\frac{14}{s^{2}}+\frac{48}{s}\right)$

Question 40

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

A) $\frac{-3}{s-5}+\frac{e^{-6 s}}{s(s-5)}$
B) $\frac{-3}{s+5}+\frac{e^{-6 s}}{s(s+5)}$
C) $-\frac{-3}{s-5}+\frac{e^{6 s}}{s(s-5)}$
D) $\frac{-3}{s+5}+\frac{e^{6 s}}{s(s+5)}$

Question 41

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

A) $\frac{e^{-9 s} \cdot 9 !}{s^{10}\left(s^{2}+9\right)}$
B) $\frac{e^{9 s} \cdot 9 !}{s^{10}\left(s^{2}+9\right)}$
C) $\frac{e^{9 s} \cdot 10 !}{s^{11}\left(s^{2}+9\right)}$
D) $\frac{e^{-9 s} \cdot 10 !}{s^{11}\left(s^{2}+9\right)}$

Question 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).

Question 43

________. Here, δ stands for the Dirac delta function.

Question 44

________. Here, δ stands for the Dirac delta function.

Question 45

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

Question 46

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

A) $\frac{5 s+20-4 e^{5 s}}{(s+2)^{2}}$
B) $\frac{5 s+20-4 e^{-5 s}}{(s+2)^{2}}$
C) $\frac{25-4 e^{5 s}}{(s+2)^{2}}$
D) $\frac{25-4 e^{-5 s}}{(s+2)^{2}}$

Question 47

Consider the following initial value problem

(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).

Question 48

Compute

A)
B)
C)
D)

Question 49

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

A) $f *\left(g^{*} h\right)=(f * g)^{*} h$
B) $f^{*} f \geq 0$
C) $f^{*} 1=f$
D) $f * g=g^{*} f$
E) $f *(g+h)=f * g+f * h$

Question 50

Compute the Laplace transform of f(t) = .

A) $\frac{1}{s}-\frac{1}{s-6}$
B) $\frac{1}{s}+\frac{1}{(s+6)^{2}}$
C) $\frac{1}{s}+\frac{1}{(s-6)^{2}}$
D) $\frac{1}{s(s-6)^{2}}$
E) $\frac{1}{s(s+6)^{2}}$
F) $-\frac{1}{s(s-6)}$

Question 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.

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) $\frac{1.6 s+1.4+F(s)}{s^{2}+6}$
B) $\frac{-1.6 s-1.4+F(s)}{s^{2}+6}$
C) $\frac{1.6 s-1.4+F(s)}{s^{2}+6}$
D) $\frac{1.4-1.6 s+F(s)}{s^{2}+6}$

Question 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.

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

Question 53

Find the function f(t) that satisfies the integral equationf(t)

Question 54

Compute * t.

Question 55

Use the convolution theorem to compute the inverse Laplace transform of

A) $\frac{t^{4}}{4 !} * e^{8 t}$
B) $\frac{t^{5}}{5 !} * e^{8 t}$
C) $\frac{t^{4}}{4 !} * e^{-8 t}$
D) $\frac{t^{5}}{5 !} * e^{-8 t}$
E) $t^{5} * e^{8 t}$
F) $t^{5} * e^{-8 t}$

Question 56

Find the function f(t) that satisfies the integral equationf(t)

Question 57

Use the convolution theorem to compute the inverse Laplace transform of
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)