9.1.1: Introduction to the Laplace Transform (Exercises)
- Page ID
- 103540
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In Exercises 1-5 find the Laplace transform from the definition: \(F(s)=\int_0^\infty e^{-st} f(t)\,dt\).
1. \(t\)
2. \(te^{-t}\)
3. \(\sinh bt\)
4. \(e^{2t}-3e^t\)
5. \(t^2\)
In Exercises 6-10 use the table of Laplace transforms to find the Laplace transforms of the following functions.
6. \(\sin^2t\)
7. \(\cos^2 2t\)
8. \(\sin t\cos t\)
9. \(\cos 2t -\cos 3t\)
10. \(\sin 2t +\cos 4t\)
In Exercises 11-14 graph the following piecewise continuous functions and evaluate \(f(t+)\), \(f(t-)\), and \(f(t)\) at each point of discontinuity.
11. \(f(t)=\left\{\begin{array}{cl} -t, & 0\le t<2,\\ t-4, & 2\le t<3,\\ 1, & t\ge 3.\end{array}\right.\)
12. \(f(t)=\left\{\begin{array}{cl} t^2+2, & 0 \le t<1,\\4, & t=1,\\ t, & t> 1.\end{array}\right.\)
13. \(f(t)=\left\{\begin{array}{rl} \sin t, & 0\le t<\pi/ 2,\\ 2\sin t, &\pi/ 2 \le t<\pi,\\ \cos t, & t\ge\pi.\end{array}\right.\)
14. \(f(t)=\left\{\begin{array}{cl}t, & 0\le t<1,\\ 2, & t=1,\\ 2-t, & 1 \le t<2,\\ 3, & t=2,\\ 6, & t> 2.\end{array}\right.\)
In Exercises 15-18 find the Laplace transform:
15. \(f(t)=\left\{\begin{array}{rl} e^{-t}, & 0\le t<1,\\ e^{-2t}, & t\ge 1.\end{array}\right.\)
16. \(f(t)=\left\{\begin{array}{rl} 1, & 0\le t< 4,\\ t, & t\ge 4.\end{array}\right.\)
17. \(f(t)=\left\{\begin{array}{rl} t, & 0\le t<1,\\ 1, & t\ge 1.\end{array}\right.\)
18. \(f(t)=\left\{\begin{array}{rl} te^t, & 0\le t<1,\\\phantom{t} e^t, & t\ge 1.\end{array}\right.\)
19. Prove that
\[\int_0^\infty e^{-st}e^{t^2} dt=\infty\nonumber \]
for every real number \(s\).
20. 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 \([u,\infty)\) (with \(u\ge 0\)) such that \(|g(t)|\le w(t)\) for \(t\ge u\) and \(\int^\infty_u w(t)\,dt\) converges. Then \(\int_0^\infty g(t)\,dt\) converges.
Use Theorem 9.1.1.1 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 \(s>s_0\).
21. Prove: If \(f\) is piecewise continuous and of exponential order then \(\lim_{s\to\infty}F(s)~=~0\).
22. Prove: If \(f\) is continuous on \([0,\infty)\) and of exponential order \(s_0>0\), then
\[{\cal L}\left(\int^t_0 f(u)\,du\right)={1\over s} {\cal L} (f), \quad s>s_0.\nonumber \] HINT: Use integration by parts to evaluate the transform on the left.
23. 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.