Math Help - Reapeated Real and Complex Roots

1. 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?

2. 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

$\chi$ $\sigma$

3. 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
$k=1+|a_1|+\cdots+|a_n|$