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Inverse Laplace transform

  1. Dec 11, 2012 #1
    1. The problem statement, all variables and given/known data
    Find inverse Laplace transform
    [tex]\mathcal {L}^{-1}[\frac{1}{(s^2+a^2)^2}][/tex]


    2. Relevant equations



    3. The attempt at a solution
    I try with theorem
    [tex]\mathcal{L}[f(t)*g(t)]=F(s)G(s)[/tex]
    So this is some multiple of
    [tex]\mathcal{L}[\sin at*\sin at][/tex]
    So [tex]\mathcal {L}^{-1}[\frac{1}{(s^2+a^2)^2}]=\propto \sin at*\sin at [/tex]
    Or
    [tex]\mathcal {L}^{-1}[\frac{1}{(s^2+a^2)^2}]=\propto \int^t_0\sin aq \sin(at-aq)dq [/tex]
    Is there some easier way?
    Tnx for answer.
     
  2. jcsd
  3. Dec 11, 2012 #2

    Ray Vickson

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    (1) Don't just say "some multiple"; give the exact factor---it matters a lot!
    (2) Just do the convolution integral; it is about as easy a way as any (other than using tables or a computer algebra package).
     
  4. Dec 11, 2012 #3
    [tex]\sin at*\sin at=\int^t_0\sin aq\sin (at-aq)dq=\int^t_0\sin aq(\sin at\cos aq-\sin aq\cos at)dq[/tex]
    So we have to solve to different integrals
    [tex]\sin at\int^t_0 \sin aq \cos aqdq=\frac{1}{2}\sin^3 at[/tex]
    and
    [tex]\cos at\int^t_0 \sin^2 aqdq=\cos at\int^t_0\frac{1-\cos 2aq}{2}dq=\frac{1}{2}t\cos at-\frac{1}{4a}\sin 2at[/tex]
    So
    [tex]\sin at*\sin at=\frac{1}{2}\sin^3 at+\frac{1}{2}t\cos at-\frac{1}{4a}\sin 2at[/tex]
    Laplace transform od ##\sin at## is ##\frac{a}{s^2+a^2}##.
    So
    [tex]\mathcal{L}[\sin at*\sin at]=\frac{a^2}{(s^2+a^2)^2}[/tex]
    So
    [tex]\mathcal{L}^{-1}[\frac{1}{(s^2+a^2)^2}]=\frac{1}{a^2}(\frac{1}{2}\sin^3 at+\frac{1}{2}t\cos at-\frac{1}{4a}\sin 2at)[/tex]
    Is that correct? Is there some easier way to do it? Tnx for the answer.
     
  5. Dec 11, 2012 #4

    Ray Vickson

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    Homework Helper

    [tex] \frac{1}{s^2+a^2} \leftrightarrow \frac{1}{a} \sin(a t)[/tex]
    so
    [tex] \frac{1}{(s^2+a^2)} \leftrightarrow \frac{1}{a^2} \int_0^t \sin(ay) \sin(a(t-y)) \, dy
    = \frac{1}{2a^3} \sin(at) -\frac{1}{2a^2} t \cos(a t).[/tex]
    I took the lazy way out and just used the computer package Maple 11. You could also use Wolfram Alpha, which is free for use.
     
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