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Math Help - Laplace of differential equation

  1. #1
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    Laplace of differential equation

    I am having trouble with the problem attached, I need to find the inverse laplace of X(s).

    I can do the 3rd term from my table of conversions directly, the 2nd term I have an answer for using partial fractions (although i'm not 100% it's correct). The 1st term is my biggest problem though.

    The answer to aim for is: x(t) = 1 - e^(2t)
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  2. #2
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    Hmm. I'm not sure I buy your derivation there. Here's the original DE:

    \ddot{x}-4\dot{x}+4x=4;\quad x(0)=0,\;\dot{x}(0)=-2.

    The LT of the DE is

    s^{2}X(s)-s\underbrace{x(0)}_{0}-\underbrace{\dot{x}(0)}_{-2}-4(sX(s)-\underbrace{x(0)}_{0})+4X(s)=\dfrac{4}{s}, or

    s^{2}X(s)+2-4sX(s)+4X(s)=\dfrac{4}{s}.

    Solving for X(s), I get

    X(s)=-\dfrac{2}{(s-2)s}.

    I think this step was incorrect:

    \mathcal{L}[4x]=\dfrac{4}{s^{2}}. It should be

    \mathcal{L}[4x]=4X(s).
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  3. #3
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    Ah yes, I see where I went wrong now, thanks!
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  4. #4
    A Plied Mathematician
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    You're welcome!
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