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Laplace transform with Convolution

  1. Feb 7, 2012 #1
    1. The problem statement, all variables and given/known data

    Use convolution theorem to solve:

    [tex]\mathfrak{L} \left \{t\int_{0}^{t} \sin \tau d\tau \right \}[/tex]

    Do not solve the integral.

    2. Relevant equations

    "Convolution Theorem" in textbook is stated as:

    [tex]\mathfrak{L}\left \{ f*g \right \}=F(s)G(s)[/tex]


    [tex]f*g=\int_{0}^{t} \ f(\tau )g(t-\tau ) d\tau[/tex]



    3. The attempt at a solution

    Not quite sure how to approach this one with convolution and not solving the integral.

    I need a t-τ instead of a t, but I can't have τ's in my Laplace because they won't go to S-space with any meaning.

    Answer in book is :

    [tex]\frac{3s^2+1}{s^2(s^2+1)^2}[/tex]


    working backwards only leads me to a partial fraction type inverse transform.

    Thanks in advance for help!
     
    Last edited: Feb 7, 2012
  2. jcsd
  3. Feb 8, 2012 #2

    vela

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    Perhaps the idea is to use the property
    $$\mathfrak{L}\{t f(t)\} = -F'(s) $$ where
    $$f(t) = \int_0^t \sin \tau \, d\tau, $$ and to find F(s) using the convolution theorem.
     
    Last edited: Feb 8, 2012
  4. Feb 8, 2012 #3
    Thanks Vela, that's exactly it
     
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