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


## Q8.1.1

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

1. $$t$$
2. $$te^{-t}$$
3. $$\sinh bt$$
4. $$e^{2t}-3e^t$$
5. $$t^2$$

2. Use the table of Laplace transforms to find the Laplace transforms of the following functions.

1. $$\cosh t\sin t$$
2. $$\sin^2t$$
3. $$\cos^2 2t$$
4. $$\cosh^2 t$$
5. $$t\sinh 2t$$
6. $$\sin t\cos t$$
7. $${\sin\left(t+{\pi\over 4}\right)}$$
8. $$\cos 2t -\cos 3t$$
9. $$\sin 2t +\cos 4t$$

3. Show that

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

for every real number $$s$$.

4. Graph the following piecewise continuous functions and evaluate $$f(t+)$$, $$f(t-)$$, and $$f(t)$$ at each point of discontinuity.

1. $$f(t)=\left\{\begin{array}{cl} -t, & 0\le t<2,\\ t-4, & 2\le t<3,\\ 1, & t\ge 3.\end{array}\right.$$
2. $$f(t)=\left\{\begin{array}{cl} t^2+2, & 0 \le t<1,\\4, & t=1,\\ t, & t> 1.\end{array}\right.$$
3. $$f(t)=\left\{\begin{array}{rl} \sin t, & 0\le t<\pi/ 2,\\ 2\sin t, &\pi/ 2 \le t<\pi,\\ \cos t, & t\ge\pi.\end{array}\right.$$
4. $$f(t)=\left\{\begin{array}{cl}t, & 0\le t<1,\\ 2, & t=1,\\ 2-t, & 1 \le t<2,\\ 3, & t=2,\\ 6, & t> 2.\end{array}\right.$$

5. Find the Laplace transform:

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

6. Prove that if $$f(t)\leftrightarrow F(s)$$ then $$t^kf(t)\leftrightarrow (-1)^kF^{(k)}(s)$$. HINT: Assume that it's permissible to differentiate the integral $$\int_{0}^{\infty}e^{-st}f(t)dt$$ with respect to $$s$$ under the integral sign.

7. Use the known Laplace transforms

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

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

8. Use the known Laplace transform $${\cal L}(1)=1/s$$ and the result of Exercise 8.1.6 to show that

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

9.

1. Show that if $$\lim_{t\to\infty} e^{-s_0t} f(t)$$ exists and is finite then $$f$$ is of exponential order $$s_0$$.
2. Show that if $$f$$ is of exponential order $$s_0$$ then $$\lim_{t \to\infty} e^{-st} f(t)=0$$ for all $$s>s_0$$.
3. 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 $$s_0$$.

10. Recall the next theorem from calculus.

Theorem $$\PageIndex{1}$$

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 $$\PageIndex{1}$$ 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 $$s>s_0$$.

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

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

${\cal L}\left(\int^t_0 f(\tau)\,d\tau\right)={1\over s} {\cal L} (f), \quad s>s_0.\nonumber$ HINT: Use integration by parts to evaluate the transform on the left.

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

${\cal L}\left({f(t)\over t}\right)=\int^\infty_s F(r)\,dr.\nonumber$ HINT: Use the results of Exercises 8.1.6 and 8.1.11.

14. Suppose $$f$$ is piecewise continuous on $$[0,\infty)$$.

1. 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 $$s>s_0$$. HINT: Use integration by parts to show that $\int_{0}^{T}e^{-st}f(t)dt = e^{-(s-s_{0})T}g(T)+(s-s_{0})\int_{0}^{T}e^{-(s-s_{0})t}g(t)dt\nonumber$
2. Show that if $${\cal L}(f)$$ exists for $$s=s_0$$ then it exists for $$s>s_0$$. Show that the function $f(t)=te^{t^2}\cos(e^{t^2})\nonumber$ has a Laplace transform defined for $$s>0$$, even though $$f$$ isn’t of exponential order.
3. Show that the function $f(t)=te^{t^2}\cos(e^{t^2})\nonumber$ has a Laplace transform defined for $$s>0$$, even though $$f$$ isn’t of exponential order.

15. Use the table of Laplace transforms and the result of Exercise 8.1.13 to find the Laplace transforms of the following functions.

1. $$\frac{\sin \omega t}{t}\quad (\omega >0)$$
2. $$\frac{\cos \omega t-1}{t}\quad (\omega >0)$$
3. $$\frac{e^{at}-e^{bt}}{t}$$
4. $$\frac{\cosh t-1}{t}$$
5. $$\frac{\sinh ^{2}t}{t}$$

16. The gamma function is defined by

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

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

1. Use integration by parts to show that $\Gamma (\alpha+1)=\alpha\Gamma (\alpha),\quad\alpha>0.\nonumber$
2. Show that $$\Gamma(n+1)=n!$$ if $$n=1$$, $$2$$, $$3$$,….
3. From (b) and the table of Laplace transforms, ${\cal L}(t^\alpha)={\Gamma (\alpha+1)\over s^{\alpha+1}},\quad s>0,\nonumber$ if $$\alpha$$ is a nonnegative integer. Show that this formula is valid for any $$\alpha>-1$$. HINT: Change the variable of integration in the integral for $$\Gamma (\alpha +1)$$.

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 $$T$$.)

1. Conclude from Theorem 8.1.6 that the Laplace transform of $$f$$ is defined for $$s>0$$.
2. Show that $F(s)={1\over 1-e^{-sT}}\int_0^T e^{-st}f(t)\,dt,\quad s>0.\nonumber$ HINT: Write $F(s)=\sum_{n=0}^{\infty}\int_{nT}^{(n+1)T}e^{-st}f(t)dt\nonumber$ Then show that $\int_{nT}^{(n+1)T}e^{-st}f(t)dt = e^{-nsT}\int_{0}^{T}e^{-st}f(t)dt\nonumber$ and recall the formula for the sum of a geometric series.

18. Use the formula given in Exercise 8.1.17b to find the Laplace transforms of the given periodic functions:

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