9.1.1: Introduction to the Laplace Transform (Exercises)

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.

Theorem 9.1.1.1

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.

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