7.1E: Introduction to the Laplace Transform (Exercises)
- Page ID
- 43323
Q7.1.1
1. Find the Laplace transforms of the following functions by evaluating the integral \(F(s)=\int_0^\infty e^{-st} f(t)\,dt\).
- \(t\)
- \(te^{-t}\)
- \(\sinh bt\)
- \(e^{2t}-3e^t\)
- \(t^2\)
- \(t^2e^{-3t}\)
- \(e^{2t}(t+1)^2\)
- \(e^{-t}\sin 4t\)
2. 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
\[\int_0^\infty e^{-st}e^{t^2} dt=\infty\nonumber \]
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.
- \(f(t)=\left\{\begin{array}{cl} -t, & 0\le t<2,\\ t-4, & 2\le t<3,\\ 1, & t\ge 3.\end{array}\right.\)
- \(f(t)=\left\{\begin{array}{cl} t^2+2, & 0 \le t<1,\\4, & t=1,\\ t, & t> 1.\end{array}\right.\)
- \(f(t)=\left\{\begin{array}{rl} \sin t, & 0\le t<\pi/ 2,\\ 2\sin t, &\pi/ 2 \le t<\pi,\\ \cos t, & t\ge\pi.\end{array}\right.\)
- \(f(t)=\left\{\begin{array}{cl}t, & 0\le t<1,\\ 2, & t=1,\\ 2-t, & 1 < t<2,\\ 3, & t=2,\\ 6, & t> 2.\end{array}\right.\)
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. 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.
- \(\frac{\sin \omega t}{t}\quad (\omega >0)\)
- \(\frac{\cos \omega t-1}{t}\quad (\omega >0)\)
- \(\frac{e^{at}-e^{bt}}{t}\)
- \(\frac{\cosh t-1}{t}\)
- \(\frac{\sinh ^{2}t}{t}\)
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 \(\alpha>0\).
- 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 \(\Gamma (\alpha +1)\).
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 \(T\).)
- Conclude from Theorem 7.1.6 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.
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.
- \( {f(t)=\left\{\begin{array}{cl} t, & 0\le t<1,\\ 2-t, & 1\le t<2,\end{array}\right.\hskip30pt f(t+2)=f(t), \quad t\ge 0}\)
- \( {f(t)=\left\{\begin{array}{rl}1, & 0\le t<{1\over 2},\\ -1, & {1\over 2}\le t<1,\end{array}\right. \hskip30pt f(t+1)=f(t),\quad t\ge 0}\)
- \(f(t)=|\sin t|\)
- \( {f(t)=\left\{\begin{array}{cl}\sin t, & 0\le t< \pi, \\ 0, &\pi\le t<2\pi,\end{array}\right.\hskip30pt f(t+2\pi)=f(t)}\)