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Math Help - Differential Equation eigenvalue!!

  1. #1
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    Differential Equation eigenvalue!!

    Hi, I am having trouble with the following problem. I dont know where I am doing it wrong... please help me out...

    Problem:

    Apply the eigenvalue method to find the particular solution to the system of differential equations



    which satifies the initial conditions







    ___________________






    Attempt:

    {A} = \begin{pmatrix}<br />
{6}&{7}\\ <br />
{5}&{8}<br />
\end{pmatrix}\\<br />
=> \left|A-\lambda I \right| = \begin{pmatrix}<br />
{6-\lambda}&{7}\\ <br />
{5}&{8-\lambda}<br />
\end{pmatrix} = 0\\<br />
=> \lambda = 13 \quad{or} \lambda = 1 \\<br />

    So I found the eigenvalues and solved for eigenvector.

    So for lambda = 13 eigenvector \begin{pmatrix}<br />
{1}\\ <br />
{1}<br />
\end{pmatrix}

    for lambda = 1 eigenvector \begin{pmatrix}<br />
{1}\\ <br />
{-5/7}<br />
\end{pmatrix}


    x(t) = {C_1}\begin{pmatrix}<br />
{1}\\ <br />
{1}<br />
\end{pmatrix}e^{13t}+{C_2}\begin{pmatrix}<br />
{1}\\ <br />
{-5/7}<br />
\end{pmatrix}e^{1t}

    Now for the given initial conditions I finally got:

    x1 = -16/3*e^(13t) - 7/3*e^(t)
    x2 = -16/3*e^(13t) + 5/3*e^(t)

    But its wrong...

    Please help...

    Many Thanks....
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  2. #2
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    Quote Originally Posted by althaemenes View Post
    Hi, I am having trouble with the following problem. I dont know where I am doing it wrong... please help me out...

    Problem:

    Apply the eigenvalue method to find the particular solution to the system of differential equations



    which satifies the initial conditions










    ___________________









    Attempt:

    {A} = \begin{pmatrix}<br />
{6}&{7}\\ <br />
{5}&{8}<br />
\end{pmatrix}\\<br />
=> \left|A-\lambda I \right| = \begin{pmatrix}<br />
{6-\lambda}&{7}\\ <br />
{5}&{8-\lambda}<br />
\end{pmatrix} = 0\\<br />
=> \lambda = 13 \quad{or} \lambda = 1 \\<br />

    So I found the eigenvalues and solved for eigenvector.

    So for lambda = 13 eigenvector \begin{pmatrix}<br />
{1}\\ <br />
{1}<br />
\end{pmatrix}

    for lambda = 1 eigenvector \begin{pmatrix}<br />
{1}\\ <br />
{-5/7}<br />
\end{pmatrix}


    x(t) = {C_1}\begin{pmatrix}<br />
{1}\\ <br />
{1}<br />
\end{pmatrix}e^{13t}+{C_2}\begin{pmatrix}<br />
{1}\\ <br />
{-5/7}<br />
\end{pmatrix}e^{1t}

    Now for the given initial conditions I finally got:

    x1 = -16/3*e^(13t) - 7/3*e^(t)
    x2 = -16/3*e^(13t) + 5/3*e^(t)

    But its wrong...

    Please help...

    Many Thanks....
    How did you arrive at your constants. I obtained

    x = - \frac{2}{3}\begin{pmatrix}<br />
{1}\\ <br />
{1}<br />
\end{pmatrix}e^{13t}- \frac{7}{3}\begin{pmatrix}<br />
{1}\\ <br />
{-5/7}<br />
\end{pmatrix}e^{t} = - \frac{2}{3}\begin{pmatrix}<br />
{1}\\ <br />
{1}<br />
\end{pmatrix}e^{13t}- \frac{1}{3}\begin{pmatrix}<br />
{7}\\ <br />
{-5}<br />
\end{pmatrix}e^{t}
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