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Math Help - Integral Eqn - laplace tranforms

  1. #1
    Junior Member
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    Dec 2008
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    Integral Eqn - laplace tranforms

    I am suppose to solve the following equation using laplace transforms and have started but am both unsure if i'm going about it the right way and where to go next,

    <br /> <br />
y(t) + \int_{0}^{t} e^{2(t - T)} . y(T) . dT = e^{2t} - t<br /> <br />
    where T is supposed to be tau.

    i re-arranged the equation to have y(t) = the rest , then used convolution theorem to say;

    <br />
y(t) = e^{2t} - t - (y * e^{2})(t)

    where (y * e^{2})(t) is the convolution

    i then took la place transforms to give;
    <br />
L(y(t)) = \frac{1}{s-2} - \frac{1}{s^{2}} - (L(y(t)) . \frac{1}{s-2}<br /> <br />
    From here i simplify but end up with terms i can't take the inverse laplace transform of in order to solve, so am thinking i must ahve gone wrong up to this point?
    Can anyone help here?

    Cheers,
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  2. #2
    Super Member
    Joined
    Aug 2008
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    Use y-hat for the transform and when I take the transform of the equation and simplify the convolution, I get: \widehat{y}+\frac{1}{s-2}\widehat{y}=\frac{1}{s-2}-\frac{1}{s^2}. Solving for yhat, I get: \widehat{y}=\frac{2}{s-1}-\frac{2}{s^2}-\frac{1}{s}. Now, you can take the inverse transform of that right?
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  3. #3
    Junior Member
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    Yes thanks!

    My problem was in simplifying to something i could take the inverse transform of, but finally got it there.

    Cheers.
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