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Laplace Transform solution help

  1. Jun 19, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the solution of the given initial value problem:
    y''+4y=upi(t)-u3pi(t) y(0)=7, y'(0)=5

    3. The attempt at a solution
    So I found the L{} of the above equation:
    s2Y-s*f(0)-f'(0)+4Y = (e-pi*s)/s-(e-3pi*s)/s

    Combining and substituting the numbers I get:
    Y=[tex]\frac{e^{-pi*s}-e^{-3pi*s}}{s(s^{2}+4)}[/tex]+[tex]\frac{6s+3}{s^2+4}[/tex]

    I know how to do the second term's inverse Laplace, but not the first. Here is what I tried:
    I can see that I can't get rid of the exponentials in any way other than using the step function again. And the other denominator factor (s^2+4) can be potentially used to get sine. So that:
    Y=[tex]\frac{1}{2}(\frac{e^{-pi*s}-e^{-3pi*s}}{s})[/tex][tex]\frac{2}{s^{2}+4}[/tex]

    This is where I don't know what to do. I can't separate them and I don't know of a way to do Laplace inverse of a product.

    Any help would be really appreciated. Thanks
     
    Last edited: Jun 19, 2009
  2. jcsd
  3. Jun 19, 2009 #2

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    partial fractions

    Also, the exponentials should be powers of s not t.
     
  4. Jun 19, 2009 #3
    How do I do partial fractions with exponentials? Do I use like Ae^(-pi*s) instead of the usual A?

    Edit: I fixed the powers
     
    Last edited: Jun 19, 2009
  5. Jun 19, 2009 #4

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    Nevermind partial fractions won't help here. You need to have proven a theorem about the laplace transform of the step function times another function.
     
  6. Jun 19, 2009 #5
    Nah, I got it, you do need partial fractions. Thanks.
     
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