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Math Help - integral equation by Laplace transforms and contour integration

  1. #1
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    integral equation by Laplace transforms and contour integration

    i have to solve \int^t_0 f(\tau)sin(t - \tau) d\tau = \cos t - \cos 3t by using Laplace transforms and then inverting it using contour integration (which i'm incredibly vague on)
    we were given the rule that if f(t) = \int^t_0 g(t - \tau)h(\tau) d\tau then F(s) = G(s)H(s), and also that the Laplace transform of an integral is L[\int^t_0 f(\tau)d\tau] = \frac{1}{s}L[f(\tau)]
    i found the Laplace transform of f(t) = \cos t is F(s) = \frac{s}{s^2 + 1} and of f(t) = \cos 3t is F(s) = \frac{s}{s^2 + 9}
    do i have to Laplace transform the integral on the left as well? or do I multiply both Laplace transforms together and then integrate by contour integration?
    i'm a little confused as to what to do next!
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  2. #2
    Grand Panjandrum
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    Re: integral equation by Laplace transforms and contour integration

    Quote Originally Posted by wik_chick88 View Post
    i have to solve \int^t_0 f(\tau)sin(t - \tau) d\tau = \cos t - \cos 3t by using Laplace transforms and then inverting it using contour integration (which i'm incredibly vague on)
    we were given the rule that if f(t) = \int^t_0 g(t - \tau)h(\tau) d\tau then F(s) = G(s)H(s), and also that the Laplace transform of an integral is L[\int^t_0 f(\tau)d\tau] = \frac{1}{s}L[f(\tau)]
    i found the Laplace transform of f(t) = \cos t is F(s) = \frac{s}{s^2 + 1} and of f(t) = \cos 3t is F(s) = \frac{s}{s^2 + 9}
    do i have to Laplace transform the integral on the left as well? or do I multiply both Laplace transforms together and then integrate by contour integration?
    i'm a little confused as to what to do next!
    Take the LT of your integal equaltion:

    \mathcal{L}\left[ \int^t_0 f(\tau)sin(t - \tau) d\tau = \cos t - \cos 3t\right]

    \left[\mathcal{L} f (s)\right] \left[ \mathcal{L} \sin(s)\right] = \left[ \mathcal{L} \cos(s)\right]- \left[ \mathcal{L} h(s)\right]

    where h(t)=\cos(3t)


    CB
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  3. #3
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    Re: integral equation by Laplace transforms and contour integration

    so am I solving for f(t)? ie. i find the Laplace transform of sin(t - \tau) and sub that and also my Laplace transforms of \cos t and \cos 3t into the equation, simplify and then inverse Laplace transform to get f(t)?

    Yes.

    CB
    Last edited by CaptainBlack; November 7th 2011 at 02:36 AM.
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