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ODE and Laplace transform

  1. Sep 13, 2012 #1
    1. The problem statement, all variables and given/known data

    I am trying to solve the following system of ODE's by Laplace transforming:
    x' = 1 + 21y - 6x \\
    y' = 6x-53y
    with the initial conditions x(0)=y(0)=0. Laplace transforming gives me (X and Y denote the Laplace transformed variables)
    sX = 1 + 21y-6x \\
    sY = 6x-53y
    From these I find
    X(s) = \frac{1}{6+s-126/(s+53)}
    The inverse Laplace transform is (I have checked this with Mathematica)
    x(t) = \frac{e^{-\frac{1}{2} \left(59+\sqrt{2713}\right) t} \left(-47+\sqrt{2713}+\left(47+\sqrt{2713}\right) e^{\sqrt{2713} t}\right)}{2 \sqrt{2713}}
    When I take t=0, then I get x(0)=1, not x(0)=0. I not quite sure where I have gone wrong, I have double-checked everything by doing it numerically too.

    Is there something that I have forgotten to do?

  2. jcsd
  3. Sep 13, 2012 #2


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    Shouldn't these be$$
    sX =\frac 1 s +21Y-6X$$ $$
    sY = 6X-53Y$$
  4. Sep 13, 2012 #3
    You are right, thanks for that! I don't know why I thought it would just be constant.

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