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From: Torsten Hennig <Torsten.Hennig@umsicht.fhg.de>
Newsgroups: comp.soft-sys.matlab
Subject: Re: Simple ODE: Linearly Damped Spring Mass System
Date: Wed, 28 Jan 2009 09:29:21 EST
Organization: The Math Forum
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> Sorry to bring this up again, but I was hoping that
> maybe anyone who is familiar with solving ODE’s
> numerically in Matlab could help me finalize the
> problem of finding the homogenous solution using
> ODE45 based on the equation below in the matrix form:
> 
> M*X"+C*X'+K*X=0 
> 

Your system can be written as a first-order system
as
X' = Y
M*Y' = -K*X-C*Y
or (if M is regular)
X' = Y
Y' = -M^(-1)*K*X-M^(-1)*C*Y
Write this system as 
Z' = A*Z
with Z=(X,Y) and A = [0,I ; -M^(-1)*K, -M^(-1)*C] 
Then (X,Y)=exp(A*t)*(X_0,Y_0) is the solution of
the homogenous equation with exp being the matrix
exponential.

Take a look at
http://www.mathworks.com/access/helpdesk/help/toolbox/symbolic/index.html?/access/helpdesk/help/toolbox/symbolic/expm.html&http://www.google.de/search?hl=de&q=MATLAB+%26+symbolic+matrix+exponential&meta= 
for the calculation of the matrix exponential.

Best wishes
Torsten.