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. \( {f(t)=\left\{\begin{array}{cl}\sin t, & 0\le t< \pi, \\[4pt] 0, &\pi\le t<2\pi,\end{array}\right.\hskip30pt f(t+2\pi)=f(t)}\)