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Thread: Question regarding a system of DE's and eigenvalues.

  1. #1
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    Question regarding a system of DE's and eigenvalues.

    Dear Colleagues,

    Could you please help me in solving the following problem:
    Show that if $\displaystyle w$ is not an eigenvalue of an $\displaystyle n\times n $ constant matrix $\displaystyle A$, then $\displaystyle x(t)$ $\displaystyle =$ $\displaystyle k$ $\displaystyle e^{wt}$ is a solution of
    $\displaystyle x^{\prime}(t)$ $\displaystyle =$ $\displaystyle A$ $\displaystyle x(t)$ - $\displaystyle b$ $\displaystyle e^{wt}$,
    where $\displaystyle b$ and $\displaystyle k$ are $\displaystyle n\times 1$ constant vectors.


    Best Regards,

    Raed.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Substitute x =k e^{ wt } in the equation. You'll obtain wk = Ak - b .

    If w is not an eigenvalue of A then, you can choose k = (A-w I )^{ -1 } b .
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