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Math Help - Operator method help

  1. #1
    Junior Member Mist's Avatar
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    Operator method help

    im trying to solve a system of differential equations of (x''(t ) + 5(x(t )) - 2y(t ) = 0
    and, -2*x(t ) + y''(t ) +2*y(t ) = 0

    but I keep getting the wrong answer..
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by Mist View Post
    im trying to solve a system of differential equations of (x''(t ) + 5(x(t )) - 2y(t ) = 0
    and, -2*x(t ) + y''(t ) +2*y(t ) = 0

    but I keep getting the wrong answer..
    Here's an outline of the solution. Write the equations as

    x'' = -5x+2y
    y'' = 2x-2y,

    or in vector form \begin{bmatrix}x''\\y''\end{bmatrix} = \begin{bmatrix}-5&2\\2&-2\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}.

    Now diagonalise that matrix A = \begin{bmatrix}-5&2\\2&-2\end{bmatrix}. You should find that the eigenvalues are –1 and –6, with corresponding eigenvectors \begin{bmatrix}1\\2\end{bmatrix} and \begin{bmatrix}-2\\1\end{bmatrix}. Let P be the matrix having those eigenvectors as columns, and let D be the diagonal matrix with the eigenvalues –1 and –6 as its diagonal elements. Then A = PDP^{-1} and so P^{-1}\begin{bmatrix}x''\\y''\end{bmatrix} = DP^{-1}\begin{bmatrix}x\\y\end{bmatrix}.

    Let \begin{bmatrix}u\\v\end{bmatrix} = P^{-1}\begin{bmatrix}x\\y\end{bmatrix}. Then u'' = -u and v'' = -6v. Those are simple harmonic motion equations, with solutions u = A\cos t + B\sin t, v = C\cos(\sqrt6t)+D\sin(\sqrt6t).

    Finally, \begin{bmatrix}x\\y\end{bmatrix} = P\begin{bmatrix}u\\v\end{bmatrix}, giving the solution

    x(t) = A\cos t + B\sin t -2(C\cos(\sqrt6t)+D\sin(\sqrt6t)),
    y(t) = 2(A\cos t + B\sin t) + C\cos(\sqrt6t)+D\sin(\sqrt6t).
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