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Math Help - Solving a System of First Order Differential Equations

  1. #1
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    Solving a System of First Order Differential Equations

    The system of first order differential equations:
     \frac{dx}{dt} = 0x - 1y
     \frac{dy}{dt} = 1x +2y
    where  x(0) = -4, y(0) = -5 has the solution  x(t) =  and  y(t) =

    When i solved for the eigenvalues, i got one double root where  \lambda = 1 . I know for a fact that given a double eigenvalue of the same value, there can only be one associated eigenvector. Therefore i must find a generalized eigenvector for the other root.

    The resulting eigenvectors are:  V_{1} = \left[\begin{array}{cc}-1\\1\end{array}\right], V_{2} = \left[\begin{array}{cc}1\\0\end{array}\right]

    I finally get to my final answer equal to:  \left[\begin{array}{cc}-5e^{t}+9te^{t}\\-4e^{t}-9te^{t}\end{array}\right] where i found the constants  C_{1} = -5 and  C_{2} = -9.

    I can't seem to get the correct answer! Anyone who contributes I thank them now.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    For

    A=\begin{bmatrix}{0}&{-1}\\{1}&{2}\end{bmatrix},\quad P=\begin{bmatrix}{-1}&{1}\\{1}&{0}\end{bmatrix},\quad J=\begin{bmatrix}{1}&{1}\\{0}&{1}\end{bmatrix}

    certainly AP=PJ or equivalently P^{-1}AP=J.

    Then, \begin{bmatrix}{x(t)}\\{y(t}\end{bmatrix}=e^{tA}\b  egin{bmatrix}{-4}\\{-5}\end{bmatrix}=Pe^{tJ}P^{-1} \begin{bmatrix}{-4}\\{-5}\end{bmatrix}=\ldots

    I ignore your possible intermediate mistake.

    Regards
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