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Elementary Differential Equations with Boundary Value Problems

Werner E. Kohler, Lee W. Johnson

Chapter 5

Laplace Transforms - all with Video Answers

Educators

Section 1

Introduction

04:23

Problem 1

As in Example 2, use the definition to find the Laplace transform for $f(t)$, if it exists. In each exercise, the given function $f(t)$ is defined on the interval $0 \leq t<\infty$. If the Laplace transform exists, give the domain of $F(s)$. In Exercises 9-12, also sketch the graph of $f(t)$.
$$
f(t)=1
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
04:23

Problem 2

As in Example 2, use the definition to find the Laplace transform for $f(t)$, if it exists. In each exercise, the given function $f(t)$ is defined on the interval $0 \leq t<\infty$. If the Laplace transform exists, give the domain of $F(s)$. In Exercises 9-12, also sketch the graph of $f(t)$.
$$
f(t)=e^{3 t}
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
04:23

Problem 3

As in Example 2, use the definition to find the Laplace transform for $f(t)$, if it exists. In each exercise, the given function $f(t)$ is defined on the interval $0 \leq t<\infty$. If the Laplace transform exists, give the domain of $F(s)$. In Exercises 9-12, also sketch the graph of $f(t)$.
$$
f(t)=t e^{-t}
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
04:23

Problem 4

As in Example 2, use the definition to find the Laplace transform for $f(t)$, if it exists. In each exercise, the given function $f(t)$ is defined on the interval $0 \leq t<\infty$. If the Laplace transform exists, give the domain of $F(s)$. In Exercises 9-12, also sketch the graph of $f(t)$.
$$
f(t)=t-5
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
04:23

Problem 5

As in Example 2, use the definition to find the Laplace transform for $f(t)$, if it exists. In each exercise, the given function $f(t)$ is defined on the interval $0 \leq t<\infty$. If the Laplace transform exists, give the domain of $F(s)$. In Exercises 9-12, also sketch the graph of $f(t)$.
$$
f(t)=t e^{t \sqrt{t}}
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
04:23

Problem 6

As in Example 2, use the definition to find the Laplace transform for $f(t)$, if it exists. In each exercise, the given function $f(t)$ is defined on the interval $0 \leq t<\infty$. If the Laplace transform exists, give the domain of $F(s)$. In Exercises 9-12, also sketch the graph of $f(t)$.
$$
f(t)=e^{(t-1)^{2}}
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
04:23

Problem 7

As in Example 2, use the definition to find the Laplace transform for $f(t)$, if it exists. In each exercise, the given function $f(t)$ is defined on the interval $0 \leq t<\infty$. If the Laplace transform exists, give the domain of $F(s)$. In Exercises 9-12, also sketch the graph of $f(t)$.
$$
f(t)=|t-1|
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
04:23

Problem 8

As in Example 2, use the definition to find the Laplace transform for $f(t)$, if it exists. In each exercise, the given function $f(t)$ is defined on the interval $0 \leq t<\infty$. If the Laplace transform exists, give the domain of $F(s)$. In Exercises 9-12, also sketch the graph of $f(t)$.
$$
f(t)=(t-2)^{2}
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
View

Problem 9

As in Example 2, use the definition to find the Laplace transform for $f(t)$, if it exists. In each exercise, the given function $f(t)$ is defined on the interval $0 \leq t<\infty$. If the Laplace transform exists, give the domain of $F(s)$. In Exercises 9-12, also sketch the graph of $f(t)$.
$$
f(t)= \begin{cases}0, & 0 \leq t<1 \\ 1, & 1 \leq t\end{cases}
$$

Suzanne W.
Suzanne W.
Numerade Educator
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Problem 10

As in Example 2, use the definition to find the Laplace transform for $f(t)$, if it exists. In each exercise, the given function $f(t)$ is defined on the interval $0 \leq t<\infty$. If the Laplace transform exists, give the domain of $F(s)$. In Exercises 9-12, also sketch the graph of $f(t)$.
$$
f(t)= \begin{cases}0, & 0 \leq t<1 \\ t-1, & 1 \leq t\end{cases}
$$

Suzanne W.
Suzanne W.
Numerade Educator
03:17

Problem 11

As in Example 2, use the definition to find the Laplace transform for $f(t)$, if it exists. In each exercise, the given function $f(t)$ is defined on the interval $0 \leq t<\infty$. If the Laplace transform exists, give the domain of $F(s)$. In Exercises 9-12, also sketch the graph of $f(t)$.
$$
f(t)= \begin{cases}0, & 0 \leq t<1 \\ 1, & 1 \leq t<2 \\ 0, & 2 \leq t\end{cases}
$$

Sean Thrasher
Sean Thrasher
Numerade Educator
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Problem 12

As in Example 2, use the definition to find the Laplace transform for $f(t)$, if it exists. In each exercise, the given function $f(t)$ is defined on the interval $0 \leq t<\infty$. If the Laplace transform exists, give the domain of $F(s)$. In Exercises 9-12, also sketch the graph of $f(t)$.
$$
f(t)= \begin{cases}0, & 0 \leq t<1 \\ t-1, & 1 \leq t<2 \\ 0, & 2<t\end{cases}
$$

Suzanne W.
Suzanne W.
Numerade Educator
03:16

Problem 13

Let $n$ be a positive integer. Using integration by parts, establish the reduction formula
$$
\int t^{n} e^{-s t} d t=-\frac{t^{n} e^{-s t}}{s}+\frac{n}{s} \int t^{n-1} e^{-s t} d t, \quad s>0 .
$$

Uma Kumari
Uma Kumari
Numerade Educator
02:02

Problem 14

For $s>0$ and $n$ a positive integer, evaluate the limits:
(a) $\lim _{t \rightarrow 0} t^{n} e^{-s t}$
(b) $\lim _{t \rightarrow \infty} t^{n} e^{-s t}$

Anurag Kumar
Anurag Kumar
Numerade Educator
11:01

Problem 15

(a) Use Exercises 13 and 14 to derive a reduction formula for the Laplace transform of $f(t)=t^{n}$,
$\mathcal{L}\left\{t^{n}\right\}=\frac{n}{s} \mathcal{L}\left\{t^{n-1}\right\}, \quad s>0 .$
(b) From Example 2, we have $\mathcal{L}\{t\}=1 / s^{2}, s>0$. Use this fact, together with reduction formula (11), to calculate $\mathcal{L}\left\{t^{k}\right\}$ for $k=2,3, \ldots, 5$.
(c) Formulate a conjecture as to the Laplace transform of $f(t)=t^{m}$, where $m$ is an arbitrary positive integer.

Aidan Mcnabb
Aidan Mcnabb
Numerade Educator
06:31

Problem 16

From a table of integrals,
$$
\begin{aligned}
&\int e^{\alpha u} \sin \beta u d u=e^{\alpha u} \frac{\alpha \sin \beta u-\beta \cos \beta u}{\alpha^{2}+\beta^{2}} \\
&\int e^{\alpha u} \cos \beta u d u=e^{\alpha u} \frac{\alpha \cos \beta u+\beta \sin \beta u}{\alpha^{2}+\beta^{2}} .
\end{aligned}
$$
Use these integrals to find the Laplace transform of $f(t)$, if it exists. If the Laplace transform exists, give the domain of $F(s)$.
$$
f(t)=\cos \omega t
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
06:57

Problem 17

From a table of integrals,
$$
\begin{aligned}
&\int e^{\alpha u} \sin \beta u d u=e^{\alpha u} \frac{\alpha \sin \beta u-\beta \cos \beta u}{\alpha^{2}+\beta^{2}} \\
&\int e^{\alpha u} \cos \beta u d u=e^{\alpha u} \frac{\alpha \cos \beta u+\beta \sin \beta u}{\alpha^{2}+\beta^{2}} .
\end{aligned}
$$
Use these integrals to find the Laplace transform of $f(t)$, if it exists. If the Laplace transform exists, give the domain of $F(s)$.
$$
f(t)=\sin \omega t
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
04:23

Problem 18

From a table of integrals,
$$
\begin{aligned}
&\int e^{\alpha u} \sin \beta u d u=e^{\alpha u} \frac{\alpha \sin \beta u-\beta \cos \beta u}{\alpha^{2}+\beta^{2}} \\
&\int e^{\alpha u} \cos \beta u d u=e^{\alpha u} \frac{\alpha \cos \beta u+\beta \sin \beta u}{\alpha^{2}+\beta^{2}} .
\end{aligned}
$$
Use these integrals to find the Laplace transform of $f(t)$, if it exists. If the Laplace transform exists, give the domain of $F(s)$.
$$
f(t)=\cos [\omega(t-2)]
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
04:23

Problem 19

From a table of integrals,
$$
\begin{aligned}
&\int e^{\alpha u} \sin \beta u d u=e^{\alpha u} \frac{\alpha \sin \beta u-\beta \cos \beta u}{\alpha^{2}+\beta^{2}} \\
&\int e^{\alpha u} \cos \beta u d u=e^{\alpha u} \frac{\alpha \cos \beta u+\beta \sin \beta u}{\alpha^{2}+\beta^{2}} .
\end{aligned}
$$
Use these integrals to find the Laplace transform of $f(t)$, if it exists. If the Laplace transform exists, give the domain of $F(s)$.
$$
f(t)=\sin [\omega(t-2)]
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
03:18

Problem 20

From a table of integrals,
$$
\begin{aligned}
&\int e^{\alpha u} \sin \beta u d u=e^{\alpha u} \frac{\alpha \sin \beta u-\beta \cos \beta u}{\alpha^{2}+\beta^{2}} \\
&\int e^{\alpha u} \cos \beta u d u=e^{\alpha u} \frac{\alpha \cos \beta u+\beta \sin \beta u}{\alpha^{2}+\beta^{2}} .
\end{aligned}
$$
Use these integrals to find the Laplace transform of $f(t)$, if it exists. If the Laplace transform exists, give the domain of $F(s)$.
$$
f(t)=e^{-t} \sin t
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
04:23

Problem 21

From a table of integrals,
$$
\begin{aligned}
&\int e^{\alpha u} \sin \beta u d u=e^{\alpha u} \frac{\alpha \sin \beta u-\beta \cos \beta u}{\alpha^{2}+\beta^{2}} \\
&\int e^{\alpha u} \cos \beta u d u=e^{\alpha u} \frac{\alpha \cos \beta u+\beta \sin \beta u}{\alpha^{2}+\beta^{2}} .
\end{aligned}
$$
Use these integrals to find the Laplace transform of $f(t)$, if it exists. If the Laplace transform exists, give the domain of $F(s)$.
$$
f(t)=e^{-2 t} \cos 4 t
$$

Amany Waheeb
Amany Waheeb
Numerade Educator
01:34

Problem 22

Use the linearity property (7) along with the transforms found in Example 2,
$$
\mathcal{L}\left\{e^{a t}\right\}=\frac{1}{s-a}, \quad s>a \quad \text { and } \quad \mathcal{L}\{t\}=\frac{1}{s^{2}}, \quad s>0,
$$
to calculate the Laplace transform $R(s)=\mathcal{L}\{r(t)\}$ of the given function $r(t)$. For what values $s$ does the Laplace transform exist?
$$
r(t)=2 e^{-5 t}+6 t
$$

Ryan Williams
Ryan Williams
Numerade Educator
01:34

Problem 23

Use the linearity property (7) along with the transforms found in Example 2,
$$
\mathcal{L}\left\{e^{a t}\right\}=\frac{1}{s-a}, \quad s>a \quad \text { and } \quad \mathcal{L}\{t\}=\frac{1}{s^{2}}, \quad s>0,
$$
to calculate the Laplace transform $R(s)=\mathcal{L}\{r(t)\}$ of the given function $r(t)$. For what values $s$ does the Laplace transform exist?
$$
r(t)=5 e^{-7 t}+t+2 e^{2 t}
$$

Ryan Williams
Ryan Williams
Numerade Educator
02:40

Problem 24

In each exercise, a function $f(t)$ is given. In Exercises 28 and 29, the symbol $\llbracket u \rrbracket$ denotes the greatest integer function, $\llbracket u \rrbracket=n$ when $n \leq u<n+1, n$ an integer; $n=\ldots,-2,-1,0,1,2, \ldots$
(a) Is $f(t)$ continuous on $0 \leq t<\infty$, discontinuous but piecewise continuous on $0 \leq t<\infty$, or neither?
(b) Is $f(t)$ exponentially bounded on $0 \leq t<\infty$ ? If so, determine values of $M$ and $a$ such that $|f(t)| \leq M e^{a t}, 0 \leq t<\infty$.
$$
f(t)=\tan t
$$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:44

Problem 25

In each exercise, a function $f(t)$ is given. In Exercises 28 and 29, the symbol $\llbracket u \rrbracket$ denotes the greatest integer function, $\llbracket u \rrbracket=n$ when $n \leq u<n+1, n$ an integer; $n=\ldots,-2,-1,0,1,2, \ldots$
(a) Is $f(t)$ continuous on $0 \leq t<\infty$, discontinuous but piecewise continuous on $0 \leq t<\infty$, or neither?
(b) Is $f(t)$ exponentially bounded on $0 \leq t<\infty$ ? If so, determine values of $M$ and $a$ such that $|f(t)| \leq M e^{a t}, 0 \leq t<\infty$.
$$
f(t)=e^{t} \sin t
$$

Arjun Singh
Arjun Singh
Numerade Educator
01:44

Problem 26

In each exercise, a function $f(t)$ is given. In Exercises 28 and 29, the symbol $\llbracket u \rrbracket$ denotes the greatest integer function, $\llbracket u \rrbracket=n$ when $n \leq u<n+1, n$ an integer; $n=\ldots,-2,-1,0,1,2, \ldots$
(a) Is $f(t)$ continuous on $0 \leq t<\infty$, discontinuous but piecewise continuous on $0 \leq t<\infty$, or neither?
(b) Is $f(t)$ exponentially bounded on $0 \leq t<\infty$ ? If so, determine values of $M$ and $a$ such that $|f(t)| \leq M e^{a t}, 0 \leq t<\infty$.
$$
f(t)=t^{2} e^{-t}
$$

Arjun Singh
Arjun Singh
Numerade Educator
01:44

Problem 27

In each exercise, a function $f(t)$ is given. In Exercises 28 and 29, the symbol $\llbracket u \rrbracket$ denotes the greatest integer function, $\llbracket u \rrbracket=n$ when $n \leq u<n+1, n$ an integer; $n=\ldots,-2,-1,0,1,2, \ldots$
(a) Is $f(t)$ continuous on $0 \leq t<\infty$, discontinuous but piecewise continuous on $0 \leq t<\infty$, or neither?
(b) Is $f(t)$ exponentially bounded on $0 \leq t<\infty$ ? If so, determine values of $M$ and $a$ such that $|f(t)| \leq M e^{a t}, 0 \leq t<\infty$.
$$
f(t)=\cosh 2 t
$$

Arjun Singh
Arjun Singh
Numerade Educator
01:44

Problem 28

In each exercise, a function $f(t)$ is given. In Exercises 28 and 29, the symbol $\llbracket u \rrbracket$ denotes the greatest integer function, $\llbracket u \rrbracket=n$ when $n \leq u<n+1, n$ an integer; $n=\ldots,-2,-1,0,1,2, \ldots$
(a) Is $f(t)$ continuous on $0 \leq t<\infty$, discontinuous but piecewise continuous on $0 \leq t<\infty$, or neither?
(b) Is $f(t)$ exponentially bounded on $0 \leq t<\infty$ ? If so, determine values of $M$ and $a$ such that $|f(t)| \leq M e^{a t}, 0 \leq t<\infty$.
$$
f(t)=\llbracket t \rrbracket
$$

Arjun Singh
Arjun Singh
Numerade Educator
01:44

Problem 29

In each exercise, a function $f(t)$ is given. In Exercises 28 and 29, the symbol $\llbracket u \rrbracket$ denotes the greatest integer function, $\llbracket u \rrbracket=n$ when $n \leq u<n+1, n$ an integer; $n=\ldots,-2,-1,0,1,2, \ldots$
(a) Is $f(t)$ continuous on $0 \leq t<\infty$, discontinuous but piecewise continuous on $0 \leq t<\infty$, or neither?
(b) Is $f(t)$ exponentially bounded on $0 \leq t<\infty$ ? If so, determine values of $M$ and $a$ such that $|f(t)| \leq M e^{a t}, 0 \leq t<\infty$.
$$
f(t)=\left[\left[e^{2 t}\right]\right]
$$

Arjun Singh
Arjun Singh
Numerade Educator
01:44

Problem 30

In each exercise, a function $f(t)$ is given. In Exercises 28 and 29, the symbol $\llbracket u \rrbracket$ denotes the greatest integer function, $\llbracket u \rrbracket=n$ when $n \leq u<n+1, n$ an integer; $n=\ldots,-2,-1,0,1,2, \ldots$
(a) Is $f(t)$ continuous on $0 \leq t<\infty$, discontinuous but piecewise continuous on $0 \leq t<\infty$, or neither?
(b) Is $f(t)$ exponentially bounded on $0 \leq t<\infty$ ? If so, determine values of $M$ and $a$ such that $|f(t)| \leq M e^{a t}, 0 \leq t<\infty$.
$$
f(t)=\frac{e^{t^{2}}}{e^{2 t}+1}
$$

Arjun Singh
Arjun Singh
Numerade Educator
01:44

Problem 31

In each exercise, a function $f(t)$ is given. In Exercises 28 and 29, the symbol $\llbracket u \rrbracket$ denotes the greatest integer function, $\llbracket u \rrbracket=n$ when $n \leq u<n+1, n$ an integer; $n=\ldots,-2,-1,0,1,2, \ldots$
(a) Is $f(t)$ continuous on $0 \leq t<\infty$, discontinuous but piecewise continuous on $0 \leq t<\infty$, or neither?
(b) Is $f(t)$ exponentially bounded on $0 \leq t<\infty$ ? If so, determine values of $M$ and $a$ such that $|f(t)| \leq M e^{a t}, 0 \leq t<\infty$.
$$
f(t)=\frac{1}{t}
$$

Arjun Singh
Arjun Singh
Numerade Educator
01:35

Problem 32

Determine whether the given improper integral converges. If the integral converges, give its value.
$$
\int_{0}^{\infty} \frac{1}{1+t^{2}} d t
$$

Aidan Mcnabb
Aidan Mcnabb
Numerade Educator
01:35

Problem 33

Determine whether the given improper integral converges. If the integral converges, give its value.
$$
\int_{0}^{\infty} \frac{t}{1+t^{2}} d t
$$

Aidan Mcnabb
Aidan Mcnabb
Numerade Educator
02:27

Problem 34

Determine whether the given improper integral converges. If the integral converges, give its value.
$$
\int_{0}^{\infty} e^{-t} \cos \left(e^{-t}\right) d t
$$

Aidan Mcnabb
Aidan Mcnabb
Numerade Educator
01:37

Problem 35

Determine whether the given improper integral converges. If the integral converges, give its value.
$$
t e^{-t^{2}} d t
$$

Aidan Mcnabb
Aidan Mcnabb
Numerade Educator
04:01

Problem 36

Suppose that $\mathcal{L}\left\{f_{1}(t)\right\}=F_{1}(s)$ and $\mathcal{L}\left\{f_{2}(t)\right\}=F_{2}(s), s>a$. Use the fact that
$$\mathcal{L}^{-1}\left\{c_{1} F_{1}(s)+c_{2} F_{2}(s)\right\}=c_{1} \mathcal{L}^{-1}\left\{F_{1}(s)\right\}+c_{2} \mathcal{L}^{-1}\left\{F_{2}(s)\right\}, \quad a<s$$
to determine the inverse Laplace transform of the given function. Refer to the examples in this section and equation (11) in Exercise 15 .
$$
F(s)=\frac{3}{s-2}
$$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
04:01

Problem 37

Suppose that $\mathcal{L}\left\{f_{1}(t)\right\}=F_{1}(s)$ and $\mathcal{L}\left\{f_{2}(t)\right\}=F_{2}(s), s>a$. Use the fact that
$$\mathcal{L}^{-1}\left\{c_{1} F_{1}(s)+c_{2} F_{2}(s)\right\}=c_{1} \mathcal{L}^{-1}\left\{F_{1}(s)\right\}+c_{2} \mathcal{L}^{-1}\left\{F_{2}(s)\right\}, \quad a<s$$
to determine the inverse Laplace transform of the given function. Refer to the examples in this section and equation (11) in Exercise 15 .
$$
F(s)=-\frac{2}{s^{2}}+\frac{1}{s+1}
$$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
04:01

Problem 38

Suppose that $\mathcal{L}\left\{f_{1}(t)\right\}=F_{1}(s)$ and $\mathcal{L}\left\{f_{2}(t)\right\}=F_{2}(s), s>a$. Use the fact that
$$\mathcal{L}^{-1}\left\{c_{1} F_{1}(s)+c_{2} F_{2}(s)\right\}=c_{1} \mathcal{L}^{-1}\left\{F_{1}(s)\right\}+c_{2} \mathcal{L}^{-1}\left\{F_{2}(s)\right\}, \quad a<s$$
to determine the inverse Laplace transform of the given function. Refer to the examples in this section and equation (11) in Exercise 15 .
$$
F(s)=\frac{4 s}{s^{2}-4}=\frac{2}{s+2}+\frac{2}{s-2}
$$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
04:01

Problem 39

Suppose that $\mathcal{L}\left\{f_{1}(t)\right\}=F_{1}(s)$ and $\mathcal{L}\left\{f_{2}(t)\right\}=F_{2}(s), s>a$. Use the fact that
$$\mathcal{L}^{-1}\left\{c_{1} F_{1}(s)+c_{2} F_{2}(s)\right\}=c_{1} \mathcal{L}^{-1}\left\{F_{1}(s)\right\}+c_{2} \mathcal{L}^{-1}\left\{F_{2}(s)\right\}, \quad a<s$$
to determine the inverse Laplace transform of the given function. Refer to the examples in this section and equation (11) in Exercise 15 .
$$
F(s)=\frac{2}{s^{2}-1}=\frac{1}{s-1}-\frac{1}{s+1}
$$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator