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Thread: Reapeated Real and Complex Roots

  1. #1
    No one in Particular VonNemo19's Avatar
    Apr 2009
    Detroit, MI

    Reapeated Real and Complex Roots


    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
    Mar 2009
    near Piacenza (Italy)
    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
    Jan 2011
    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

    Last edited by LoblawsLawBlog; Mar 3rd 2011 at 08:22 PM.
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