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Mathematics LibreTexts

8.1E: Introduction to the Laplace Transform (Exercises)

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    18292
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    [exer:8.1.1] Find the Laplace transforms of the following functions by evaluating the integral \(F(s)=\int_0^\infty e^{-st} f(t)\,dt\).

    a

    \(t\) &

    b

    \(te^{-t}\) &

    c

    \(\sinh bt\)

    d

    \(e^{2t}-3e^t\) &

    e

    \(t^2\)

    [exer:8.1.2] Use the table of Laplace transforms to find the Laplace transforms of the following functions.

    a

    \(\cosh t\sin t\) &

    b

    \(\sin^2t\) &

    c

    \(\cos^2 2t\)

    d

    \(\cosh^2 t\) &

    e

    \(t\sinh 2t\) &

    f

    \(\sin t\cos t\)

    g

    \( {\sin\left(t+{\pi\over 4}\right)}\) &

    h

    \(\cos 2t -\cos 3t\) &

    i

    \(\sin 2t +\cos 4t\)

    [exer:8.1.3] Show that

    \[\int_0^\infty e^{-st}e^{t^2} dt=\infty\]

    for every real number \(s\).

    [exer:8.1.4] Graph the following piecewise continuous functions and evaluate \(f(t+)\), \(f(t-)\), and \(f(t)\) at each point of discontinuity.

    @p168pt@p168pt

    a

    \(f(t)=\left\{\begin{array}{cl} -t, & 0\le t<2,\\ t-4, & 2\le t<3,\\ 1, & t\ge 3.\end{array}\right.\) &

    b

    \(f(t)=\left\{\begin{array}{cl} t^2+2, & 0 \le t<1,\\4, & t=1,\\ t, & t> 1.\end{array}\right.\)

    c

    \(f(t)=\left\{\begin{array}{rl} \sin t, & 0\le t<\pi/ 2,\\ 2\sin t, &\pi/ 2 \le t<\pi,\\ \cos t, & t\ge\pi.\end{array}\right.\) &

    d

    \(f(t)=\left\{\begin{array}{cl}t, & 0\le t<1,\\ 2, & t=1,\\ 2-t, & 1 \le t<2,\\ 3, & t=2,\\ 6, & t> 2.\end{array}\right.\)

    [exer:8.1.5] Find the Laplace transform:

    @p168pt@p168pt

    a

    \(f(t)=\left\{\begin{array}{rl} e^{-t}, & 0\le t<1,\\ e^{-2t}, & t\ge 1.\end{array}\right.\) &

    b

    \(f(t)=\left\{\begin{array}{rl} 1, & 0\le t< 4,\\ t, & t\ge 4.\end{array}\right.\)

    c

    \(f(t)=\left\{\begin{array}{rl} t, & 0\le t<1,\\ 1, & t\ge 1.\end{array}\right.\)&

    d

    \(f(t)=\left\{\begin{array}{rl} te^t, & 0\le t<1,\\\phantom{t} e^t, & t\ge 1.\end{array}\right.\)

    [exer:8.1.6] Prove that if \(f(t)\leftrightarrow F(s)\) then \(t^kf(t)\leftrightarrow (-1)^kF^{(k)}(s)\).

    [exer:8.1.7] Use the known Laplace transforms

    \[{\cal L}(e^{\lambda t}\sin\omega t)={\omega\over(s-\lambda)^2+\omega^2} \quad\mbox{and }\quad {\cal L}(e^{\lambda t}\cos\omega t)={s-\lambda\over(s-\lambda)^2+\omega^2}\]

    and the result of Exercise [exer:8.1.6} to find \({\cal L}(te^{\lambda t}\cos\omega t)\) and \({\cal L}(te^{\lambda t}\sin\omega t)\).

    [exer:8.1.8] Use the known Laplace transform \({\cal L}(1)=1/s\) and the result of Exercise [exer:8.1.6} to show that

    \[{\cal L}(t^n)={n!\over s^{n+1}},\quad n=\mbox{ integer}.\]

    [exer:8.1.9]

    Show that if \(\lim_{t\to\infty} e^{-s_0t} f(t)\) exists and is finite then \(f\) is of exponential order \(s_0\).

    Show that if \(f\) is of exponential order \(s_0\) then \(\lim_{t \to\infty} e^{-st} f(t)=0\) 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 \(s_0\).

    [exer:8.1.10] 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 \(\int^\infty_\tau w(t)\,dt\) converges. Then \(\int_0^\infty g(t)\,dt\) converges.

    Use Theorem A 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 \(s>s_0\).

    [exer:8.1.11] Prove: If \(f\) is piecewise continuous and of exponential order then \(\lim_{s\to\infty}F(s)~=~0\).

    [exer:8.1.12] Prove: If \(f\) is continuous on \([0,\infty)\) and of exponential order \(s_0>0\), then

    \[{\cal L}\left(\int^t_0 f(\tau)\,d\tau\right)={1\over s} {\cal L} (f), \quad s>s_0.\]

    [exer:8.1.13] Suppose \(f\) is piecewise continuous and of exponential order, and that \(\lim_{t\to 0+} f(t)/t\) exists. Show that

    \[{\cal L}\left({f(t)\over t}\right)=\int^\infty_s F(r)\,dr.\]

    [exer:8.1.14] Suppose \(f\) is piecewise continuous on \([0,\infty)\).

    Prove: If the integral \(g(t)=\int^t_0 e^{-s_0\tau} f(\tau)\,d\tau\) satisfies the inequality \(|g(t)|\le M\; (t\ge 0)\), then \(f\) has a Laplace transform \(F(s)\) defined for \(s>s_0\).

    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.

    [exer:8.1.15] Use the table of Laplace transforms and the result of Exercise [exer:8.1.13} to find the Laplace transforms of the following functions.

    a

    \( {{\sin\omega t\over t}\quad(\omega>0)}\) &

    b

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    c

    \(

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    d

    \(

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    \) &

    e

    \(

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    \)

    [exer:8.1.16] The gamma function is defined by

    \[\Gamma (\alpha)=\int_0^\infty x^{\alpha-1}e^{-x}\,dx,\]

    which can be shown to converge if \(\alpha>0\).

    Use integration by parts to show that

    \[\Gamma (\alpha+1)=\alpha\Gamma (\alpha),\quad\alpha>0.\]

    Show that \(\Gamma(n+1)=n!\) if \(n=1\), \(2\), \(3\),….

    From

    b

    and the table of Laplace transforms,

    \[{\cal L}(t^\alpha)={\Gamma (\alpha+1)\over s^{\alpha+1}},\quad s>0,\]

    if \(\alpha\) is a nonnegative integer. Show that this formula is valid for any \(\alpha>-1\).

    [exer:8.1.17] 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 [thmtype:8.1.6} that the Laplace transform of \(f\) is defined for \(s>0\).

    b

    Show that

    \[F(s)={1\over 1-e^{-sT}}\int_0^T e^{-st}f(t)\,dt,\quad s>0.\]

    [exer:8.1.18] Use the formula given in Exercise [exer:8.1.17}

    b

    to find the Laplace transforms of the given periodic functions:

    \( {f(t)=\left\{\begin{array}{cl} t, & 0\le t<1,\\ 2-t, & 1\le t<2,\end{array}\right.\hskip30pt f(t+2)=f(t), \quad t\ge 0}\)

    \( {f(t)=\left\{\begin{array}{rl}1, & 0\le t<{1\over 2},\\ -1, & {1\over 2}\le t<1,\end{array}\right. \hskip30pt f(t+1)=f(t),\quad t\ge 0}\)

    \(f(t)=|\sin t|\)

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