
# 8.1E: Introduction to the Laplace Transform (Exercises)

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[exer:8.1.1] Find the Laplace transforms of the following functions by evaluating the integral $$F(s)=\int_0^\infty e^{-st} f(t)\,dt$$.

## a

$$t$$ &

## b

$$te^{-t}$$ &

## c

$$\sinh bt$$

## d

$$e^{2t}-3e^t$$ &

## e

$$t^2$$

[exer:8.1.2] Use the table of Laplace transforms to find the Laplace transforms of the following functions.

## a

$$\cosh t\sin t$$ &

## b

$$\sin^2t$$ &

## c

$$\cos^2 2t$$

## d

$$\cosh^2 t$$ &

## e

$$t\sinh 2t$$ &

## f

$$\sin t\cos t$$

## g

$${\sin\left(t+{\pi\over 4}\right)}$$ &

## h

$$\cos 2t -\cos 3t$$ &

## i

$$\sin 2t +\cos 4t$$

[exer:8.1.3] Show that

$\int_0^\infty e^{-st}e^{t^2} dt=\infty$

for every real number $$s$$.

[exer:8.1.4] Graph the following piecewise continuous functions and evaluate $$f(t+)$$, $$f(t-)$$, and $$f(t)$$ at each point of discontinuity.

@p168pt@p168pt

## a

$$f(t)=\left\{\begin{array}{cl} -t, & 0\le t<2,\\ t-4, & 2\le t<3,\\ 1, & t\ge 3.\end{array}\right.$$ &

## b

$$f(t)=\left\{\begin{array}{cl} t^2+2, & 0 \le t<1,\\4, & t=1,\\ t, & t> 1.\end{array}\right.$$

## c

$$f(t)=\left\{\begin{array}{rl} \sin t, & 0\le t<\pi/ 2,\\ 2\sin t, &\pi/ 2 \le t<\pi,\\ \cos t, & t\ge\pi.\end{array}\right.$$ &

## d

$$f(t)=\left\{\begin{array}{cl}t, & 0\le t<1,\\ 2, & t=1,\\ 2-t, & 1 \le t<2,\\ 3, & t=2,\\ 6, & t> 2.\end{array}\right.$$

[exer:8.1.5] Find the Laplace transform:

@p168pt@p168pt

## a

$$f(t)=\left\{\begin{array}{rl} e^{-t}, & 0\le t<1,\\ e^{-2t}, & t\ge 1.\end{array}\right.$$ &

## b

$$f(t)=\left\{\begin{array}{rl} 1, & 0\le t< 4,\\ t, & t\ge 4.\end{array}\right.$$

## c

$$f(t)=\left\{\begin{array}{rl} t, & 0\le t<1,\\ 1, & t\ge 1.\end{array}\right.$$&

## d

$$f(t)=\left\{\begin{array}{rl} te^t, & 0\le t<1,\\\phantom{t} e^t, & t\ge 1.\end{array}\right.$$

[exer:8.1.6] Prove that if $$f(t)\leftrightarrow F(s)$$ then $$t^kf(t)\leftrightarrow (-1)^kF^{(k)}(s)$$.

[exer:8.1.7] Use the known Laplace transforms

${\cal L}(e^{\lambda t}\sin\omega t)={\omega\over(s-\lambda)^2+\omega^2} \quad\mbox{and }\quad {\cal L}(e^{\lambda t}\cos\omega t)={s-\lambda\over(s-\lambda)^2+\omega^2}$

and the result of Exercise [exer:8.1.6} to find $${\cal L}(te^{\lambda t}\cos\omega t)$$ and $${\cal L}(te^{\lambda t}\sin\omega t)$$.

[exer:8.1.8] Use the known Laplace transform $${\cal L}(1)=1/s$$ and the result of Exercise [exer:8.1.6} to show that

${\cal L}(t^n)={n!\over s^{n+1}},\quad n=\mbox{ integer}.$

[exer:8.1.9]

Show that if $$\lim_{t\to\infty} e^{-s_0t} f(t)$$ exists and is finite then $$f$$ is of exponential order $$s_0$$.

Show that if $$f$$ is of exponential order $$s_0$$ then $$\lim_{t \to\infty} e^{-st} f(t)=0$$ for all $$s>s_0$$.

Show that if $$f$$ is of exponential order $$s_0$$ and $$g(t)=f(t+\tau)$$ where $$\tau>0$$, then $$g$$ is also of exponential order $$s_0$$.

[exer:8.1.10] Recall the next theorem from calculus.

Let $$g$$ be integrable on $$[0,T]$$ for every $$T>0.$$ Suppose there’s a function $$w$$ defined on some interval $$[\tau,\infty)$$ (with $$\tau\ge 0$$) such that $$|g(t)|\le w(t)$$ for $$t\ge\tau$$ and $$\int^\infty_\tau w(t)\,dt$$ converges. Then $$\int_0^\infty g(t)\,dt$$ converges.

Use Theorem A to show that if $$f$$ is piecewise continuous on $$[0,\infty)$$ and of exponential order $$s_0$$, then $$f$$ has a Laplace transform $$F(s)$$ defined for $$s>s_0$$.

[exer:8.1.11] Prove: If $$f$$ is piecewise continuous and of exponential order then $$\lim_{s\to\infty}F(s)~=~0$$.

[exer:8.1.12] Prove: If $$f$$ is continuous on $$[0,\infty)$$ and of exponential order $$s_0>0$$, then

${\cal L}\left(\int^t_0 f(\tau)\,d\tau\right)={1\over s} {\cal L} (f), \quad s>s_0.$

[exer:8.1.13] Suppose $$f$$ is piecewise continuous and of exponential order, and that $$\lim_{t\to 0+} f(t)/t$$ exists. Show that

${\cal L}\left({f(t)\over t}\right)=\int^\infty_s F(r)\,dr.$

[exer:8.1.14] Suppose $$f$$ is piecewise continuous on $$[0,\infty)$$.

Prove: If the integral $$g(t)=\int^t_0 e^{-s_0\tau} f(\tau)\,d\tau$$ satisfies the inequality $$|g(t)|\le M\; (t\ge 0)$$, then $$f$$ has a Laplace transform $$F(s)$$ defined for $$s>s_0$$.

Show that if $${\cal L}(f)$$ exists for $$s=s_0$$ then it exists for $$s>s_0$$. Show that the function

$f(t)=te^{t^2}\cos(e^{t^2})$

has a Laplace transform defined for $$s>0$$, even though $$f$$ isn’t of exponential order.

Show that the function

$f(t)=te^{t^2}\cos(e^{t^2})$

has a Laplace transform defined for $$s>0$$, even though $$f$$ isn’t of exponential order.

[exer:8.1.15] Use the table of Laplace transforms and the result of Exercise [exer:8.1.13} to find the Laplace transforms of the following functions.

## a

$${{\sin\omega t\over t}\quad(\omega>0)}$$ &

## b

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## c

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## d

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## e

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[exer:8.1.16] The gamma function is defined by

$\Gamma (\alpha)=\int_0^\infty x^{\alpha-1}e^{-x}\,dx,$

which can be shown to converge if $$\alpha>0$$.

Use integration by parts to show that

$\Gamma (\alpha+1)=\alpha\Gamma (\alpha),\quad\alpha>0.$

Show that $$\Gamma(n+1)=n!$$ if $$n=1$$, $$2$$, $$3$$,….

From

## b

and the table of Laplace transforms,

${\cal L}(t^\alpha)={\Gamma (\alpha+1)\over s^{\alpha+1}},\quad s>0,$

if $$\alpha$$ is a nonnegative integer. Show that this formula is valid for any $$\alpha>-1$$.

[exer:8.1.17] Suppose $$f$$ is continuous on $$[0, T]$$ and $$f(t+T)=f(t)$$ for all $$t\ge 0$$. (We say in this case that $$f$$ is periodic with period $$T$$.)

Conclude from Theorem [thmtype:8.1.6} that the Laplace transform of $$f$$ is defined for $$s>0$$.

## b

Show that

$F(s)={1\over 1-e^{-sT}}\int_0^T e^{-st}f(t)\,dt,\quad s>0.$

[exer:8.1.18] Use the formula given in Exercise [exer:8.1.17}

## b

to find the Laplace transforms of the given periodic functions:

$${f(t)=\left\{\begin{array}{cl} t, & 0\le t<1,\\ 2-t, & 1\le t<2,\end{array}\right.\hskip30pt f(t+2)=f(t), \quad t\ge 0}$$

$${f(t)=\left\{\begin{array}{rl}1, & 0\le t<{1\over 2},\\ -1, & {1\over 2}\le t<1,\end{array}\right. \hskip30pt f(t+1)=f(t),\quad t\ge 0}$$

$$f(t)=|\sin t|$$

$${f(t)=\left\{\begin{array}{cl}\sin t, & 0\le t< \pi, \\ 0, &\pi\le t<2\pi,\end{array}\right.\hskip30pt f(t+2\pi)=f(t)}$$