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Solve this system of linear ODEs?

  1. Jul 1, 2010 #1
    1. The problem statement, all variables and given/known data

    Solve this system of linear ODEs:

    1) x''(t) = x + y
    2) y''(t) = x + y

    Just fyi, this is part of a much larger problem but I need to solve this system!

    2. Relevant equations

    See above.

    3. The attempt at a solution

    Okay so I think the most logical way to solve this would be to set x''(t) = y''(t).

    x''(t) = y''(t)
    x' = y' + c1
    x = y + c1t + c2

    which implies

    3) x'' = 2x - c1t - c2
    4) y'' = 2y + c1t + c2

    But I am not sure what to do from here. Apparently there should be 4 arbitrary constants in the final answer. But wouldn't solving x'' for x give you 2 and then solving y'' for y give you two more?

    Thanks for any help.
  2. jcsd
  3. Jul 1, 2010 #2


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    Gold Member

    Did you find the eigenvalues and eigenvectors? What is the general solution to this linear system of ODEs? Yes you will end up with 4 constants but thats the minor step
  4. Jul 1, 2010 #3
    You could also write it as a vector equation
    \vec x''(t) = \left(\begin{array}{cc} 1&1\\1&1\end{array}\right)\vec x(t)
    and go on from there.
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