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Using Laplace Transforms to Solve DE

  1. Dec 7, 2008 #1
    1. The problem statement, all variables and given/known data
    The problem is this:

    [tex]y'' - 4y = e^{-t} , y(0) = 1, y'(0) = 0 [/tex]


    2. Relevant equations
    L{y(t)} = Y(s)
    L{y'(t)} = sY(s) - y(0)
    L{y''(t)} = s^2Y(s) - sy(0) - y(0)


    3. The attempt at a solution

    Ok, so I plugged the Laplace transforms for y'' and y into the equation as well as for e^(-t) and got:

    [tex]Y(s) = \frac{1}{s(s-4)(s+1)} - \frac{1}{(s-4)} - \frac{4}{4(s+1)}[/tex]

    From that point on I would need to perform a partial fraction decomposition on the first term and last term, the middle term is ok. Doing so I get:

    [tex]Y(s) = \frac{-1}{4s} + \frac{-1}{20(s-4)} + \frac{1}{5(s+1)}- \frac{1}{(s-1)} + \frac{1}{s} - \frac{1}{(s-4)}[/tex]

    However, the book is getting answers with e^(2t). I am not seeing how anything I have in that last line will yield that, so I assume I am doing something wrong.
     
  2. jcsd
  3. Dec 7, 2008 #2

    rock.freak667

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    Homework Helper

    Check over the transform of the left side. I believe you are supposed to get

    s2Y(s)-sY(0)-Y'(0)-4Y(s)
     
  4. Dec 7, 2008 #3
    ...It's stupid mistakes like the one I made above that are frustrating. Thank you.
     
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