# Reapeated Real and Complex Roots

• Mar 2nd 2011, 09:40 AM
VonNemo19
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 $\displaystyle m_1=m_2=...=m_n$, then the general solution is given by the linear

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

I know how to show this for $\displaystyle n=2$, but the general case I'm having a little trouble with. Any help?
• Mar 2nd 2011, 12:16 PM
chisigma
Let's suppose that the characteristic equation of the linear homogeneous DE has a real root m of multeplicity n, so that is...

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

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

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

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Mar 3rd 2011, 07:57 PM
LoblawsLawBlog
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

$\displaystyle ||\phi(x)||=[|\phi(x)|^2+\cdots+|\phi^{(n-1)}(x)|^2]^{1/2}$
and
$\displaystyle k=1+|a_1|+\cdots+|a_n|$