7.1E: Introduction to the Laplace Transform (Exercises)

Q7.1.1

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. $t^2$
6. $t^2e^{-3t}$
7. $e^{2t}(t+1)^2$
8. $e^{-t}\sin 4t$

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

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. 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 $s>s_0$.

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

8. 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.

9. 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.

10. Use the result of Exercise 7.1.9 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. $\frac{\sinh ^{2}t}{t}$

11. 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 $\Gamma (\alpha +1)$.

12. 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 7.1.6 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.

13. Use the formula given in Exercise 7.1.12b to find the Laplace transforms of the given periodic functions. Graph each function over one period.

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