8.1E: Introduction to the Laplace Transform (Exercises)
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Q8.1.1
1. Find the Laplace transforms of the following functions by evaluating the integral \(\displaystyle F(s)=\int_0^\infty e^{-st} f(t)\,dt\).
- \(t\)
- \(te^{-t}\)
- \(\sinh bt\)
- \(e^{2t}-3e^t\)
- \(t^2\)
2. Use the table of Laplace transforms to find the Laplace transforms of the following functions.
- \(\cosh t\sin t\)
- \(\sin^2t\)
- \(\cos^2 2t\)
- \(\cosh^2 t\)
- \(t\sinh 2t\)
- \(\sin t\cos t\)
- \( {\sin\left(t+{\pi\over 4}\right)}\)
- \(\cos 2t -\cos 3t\)
- \(\sin 2t +\cos 4t\)
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.
- \(f(t)=\left\{\begin{array}{cl} -t, & 0\le t<2,\\ t-4, & 2\le t<3,\\ 1, & t\ge 3.\end{array}\right.\)
- \(f(t)=\left\{\begin{array}{cl} t^2+2, & 0 \le t<1,\\4, & t=1,\\ t, & t> 1.\end{array}\right.\)
- \(f(t)=\left\{\begin{array}{rl} \sin t, & 0\le t<\pi/ 2,\\ 2\sin t, &\pi/ 2 \le t<\pi,\\ \cos t, & t\ge\pi.\end{array}\right.\)
- \(f(t)=\left\{\begin{array}{cl}t, & 0\le t<1,\\ 2, & t=1,\\ 2-t, & 1 \le t<2,\\ 3, & t=2,\\ 6, & t> 2.\end{array}\right.\)
5. Find the Laplace transform:
- \(f(t)=\left\{\begin{array}{rl} e^{-t}, & 0\le t<1,\\ e^{-2t}, & t\ge 1.\end{array}\right.\)
- \(f(t)=\left\{\begin{array}{rl} 1, & 0\le t< 4,\\ t, & t\ge 4.\end{array}\right.\)
- \(f(t)=\left\{\begin{array}{rl} t, & 0\le t<1,\\ 1, & t\ge 1.\end{array}\right.\)
- \(f(t)=\left\{\begin{array}{rl} te^t, & 0\le t<1,\\\phantom{t} e^t, & t\ge 1.\end{array}\right.\)
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 \(\displaystyle \int_{0}^{\infty}e^{-st}f(t)dt\) 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:
- Show that if \(\displaystyle \lim_{t\to\infty} e^{-s_0t} f(t)\) exists and is finite then \(f\) is of exponential order \(s_0\).
- Show that if \(f\) is of exponential order \(s_0\) then \(\displaystyle \lim_{t \to\infty} e^{-st} f(t)=0\) for all \(s>s_0\).
- 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.
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 \([0,\infty)\).
- 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 \[\int_{0}^{T}e^{-st}f(t)dt = e^{-(s-s_{0})T}g(T)+(s-s_{0})\int_{0}^{T}e^{-(s-s_{0})t}g(t)dt\nonumber \]
- 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.
15. Use the table of Laplace transforms and the result of Exercise 8.1.13 to find the Laplace transforms of the following functions.
- \(\dfrac{\sin \omega t}{t}\quad (\omega >0)\)
- \(\dfrac{\cos \omega t-1}{t}\quad (\omega >0)\)
- \(\dfrac{e^{at}-e^{bt}}{t}\)
- \(\dfrac{\cosh t-1}{t}\)
- \(\dfrac{\sinh ^{2}t}{t}\)
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 \(\alpha>0\).
- Use integration by parts to show that \[\Gamma (\alpha+1)=\alpha\Gamma (\alpha),\quad\alpha>0.\nonumber \]
- Show that \(\Gamma(n+1)=n!\) if \(n=1\), \(2\), \(3\),….
- From (b) and the table of Laplace transforms, \[{\cal L}(t^\alpha)={\Gamma (\alpha+1)\over s^{\alpha+1}},\quad s>0,\nonumber \] 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 \(T\).)
- Conclude from Theorem 8.1.6 that the Laplace transform of \(f\) is defined for \(s>0\).
- 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:
- \( {f(t)=\left\{\begin{array}{cl} t, & 0\le t<1,\\ 2-t, & 1\le t<2,\end{array}\right.\hskip30pt f(t+2)=f(t), \quad t\ge 0}\)
- \( {f(t)=\left\{\begin{array}{rl}1, & 0\le t<{1\over 2},\\ -1, & {1\over 2}\le t<1,\end{array}\right. \hskip30pt f(t+1)=f(t),\quad t\ge 0}\)
- \(f(t)=|\sin t|\)
- \( {f(t)=\left\{\begin{array}{cl}\sin t, & 0\le t< \pi, \\ 0, &\pi\le t<2\pi,\end{array}\right.\hskip30pt f(t+2\pi)=f(t)}\)