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Thread: Differential Equation eigenvalue!!

  1. #1
    Junior Member
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    Mar 2009
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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:

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

    So I found the eigenvalues and solved for eigenvector.

    So for lambda = 13 eigenvector $\displaystyle \begin{pmatrix}
    {1}\\
    {1}
    \end{pmatrix}$

    for lambda = 1 eigenvector $\displaystyle \begin{pmatrix}
    {1}\\
    {-5/7}
    \end{pmatrix}$


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

    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
    MHF Contributor
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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:

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

    So I found the eigenvalues and solved for eigenvector.

    So for lambda = 13 eigenvector $\displaystyle \begin{pmatrix}
    {1}\\
    {1}
    \end{pmatrix}$

    for lambda = 1 eigenvector $\displaystyle \begin{pmatrix}
    {1}\\
    {-5/7}
    \end{pmatrix}$


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

    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

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