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Math Help - Inverse Laplace using partial fractions

  1. #1
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    Inverse Laplace using partial fractions

    So the question is an initial value problem involving a second order dif eq and we need to then use inverse Laplace.

    y''+4y= cost y(0)=1 and y'(0)=0

    expanding,

    s^2Y(s) - sy(0) -y'(0) + 4Y(s) = \frac{s}{s^2 +1}

    Then solve for Y(s)

     Y(s) = \frac{s}{(s^2 +4)(s^2+1)} - \frac{s}{s^2+4}

    the term on the right gives us cos2t

    for the term on the left I used Partial fractions. our professor doesn't want us to solve for A,B,C etc. he just wants the functions. So following his example in class I did the following:

     \frac{As+B}{(s^2+1)} + \frac{Cs+D}{s^2+4}<br />
\Rightarrow \frac{As}{s^2+1} + \frac{B}{s^2+1} + \frac{Cs}{s^2+4}+\frac{D}{s^2+4}<br />

    Using the table I got
    Y(s) = Acost + Bsint + (C-1)cos2t + Dsin2t

    The answer in the back of the book is
    \frac{1}{3}cos(2t) + \frac{2}{3}cos(t)

    Can anyone tell me what I'm doing wrong? I basically did EXACTLY what he did in class.

    Thanks so much!
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  2. #2
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    shouldn't Y(s) be +s/(s^2+4) rather than minus. You add s to the other side not subtract it.
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  3. #3
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    Yes I'm sorry that should be a plus!
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  4. #4
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    What values did you find for A, B, C, and D?
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