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Math Help - Reapeated Real and Complex Roots

  1. #1
    No one in Particular VonNemo19's Avatar
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    Reapeated Real and Complex Roots

    Hi.

    My book on the subject of differential equations tends to omit things from time to time, and one such thing - namely, how it is that we can show that if an auxiliary equation to an nth order DE with constant

    coeffiecients has as its roots m_1=m_2=...=m_n, then the general solution is given by the linear

    combination of the terms e^{mx},x^{}e^{mx},x^{2}e^{mx},...,x^{n-1}e^{mx}.

    I know how to show this for n=2, but the general case I'm having a little trouble with. Any help?
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  2. #2
    MHF Contributor chisigma's Avatar
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    Let's suppose that the characteristic equation of the linear homogeneous DE has a real root m of multeplicity n, so that is...

    \displaystyle Y(s)= \sum_{k=1}^{n} \frac{c_{k}}{(s-m)^{k}} (1)

    In that case the solution of the DE is...

    \displaystyle y(t)= \mathcal {L}^{-1} \{Y(s)\} = \sum_{k=1}^{n} \frac{c_{k}}{(k-1)!}\ t^{k-1}\ e^{m\ t} (2)

    Kind regards

    \chi \sigma
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  3. #3
    Junior Member
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    In addition to chisigma's post, I think this book explains it well starting around page 71. The preview might end before it explains everything though. In particular, it ends a little before the proof of the uniqueness theorem, which is crucial. I don't really know any simpler proofs than that and I can post a sketch later if you want. Hopefully you can follow chisigma's and save me some writing

    An introduction to ordinary ... - Google Books

    edit: I just saw that around page 54 he proves the theorem for n=2. The proof carries over to the general case if you change it to

    ||\phi(x)||=[|\phi(x)|^2+\cdots+|\phi^{(n-1)}(x)|^2]^{1/2}
    and
    k=1+|a_1|+\cdots+|a_n|
    Last edited by LoblawsLawBlog; March 3rd 2011 at 09:22 PM.
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