8.1E: Introduction to the Laplace Transform (Exercises)

Q8.1.1

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

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 $\displaystyle \int_0^\infty e^{-st}e^{t^2} dt=\infty$ 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. $f(t)=\left\{\begin{array}{cl} -t, & 0\le t<2,\\[4pt] t-4, & 2\le t<3,\\[4pt] 1, & t\ge 3.\end{array}\right.$
2. $f(t)=\left\{\begin{array}{cl} t^2+2, & 0 \le t<1,\\[4pt]4, & t=1,\\[4pt] t, & t> 1.\end{array}\right.$
3. $f(t)=\left\{\begin{array}{rl} \sin t, & 0\le t<\pi/ 2,\\[4pt] 2\sin t, &\pi/ 2 \le t<\pi,\\[4pt] \cos t, & t\ge\pi.\end{array}\right.$
4. $f(t)=\left\{\begin{array}{cl}t, & 0\le t<1,\\[4pt] 2, & t=1,\\[4pt] 2-t, & 1 \le t<2,\\[4pt] 3, & t=2,\\[4pt] 6, & t> 2.\end{array}\right.$

5. Find the Laplace transform:

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

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 $\displaystyle \int_{0}^{\infty}e^{-st}f(t)dt$ 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. Exponential order:

1. Show that if $\displaystyle \lim_{t\to\infty} e^{-s_0t} f(t)$ exists and is finite then $f$ is of exponential order $s_0$.
2. Show that if $f$ is of exponential order $s_0$ then $\displaystyle \lim_{t \to\infty} e^{-st} f(t)=0$ 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 8.1E.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 $\displaystyle \int^\infty_\tau w(t)\,dt$ converges. Then $\displaystyle \int_0^\infty g(t)\,dt$ converges.

Use Theorem 8.1E.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 $\displaystyle \lim_{s\to\infty}F(s)~=~0$.

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 $\displaystyle \lim_{t\to 0+} f(t)/t$ 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 $\displaystyle g(t)=\int^t_0 e^{-s_0\tau} f(\tau)\,d\tau$ 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 $f(t)=te^{t^2}\cos(e^{t^2})$ has a Laplace transform defined for $s>0$, even though $f$ isn’t of exponential order.
3. Show that the function $f(t)=te^{t^2}\cos(e^{t^2})$ 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. $\dfrac{\sin \omega t}{t}\quad (\omega >0)$
2. $\dfrac{\cos \omega t-1}{t}\quad (\omega >0)$
3. $\dfrac{e^{at}-e^{bt}}{t}$
4. $\dfrac{\cosh t-1}{t}$
5. $\dfrac{\sinh ^{2}t}{t}$

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