Forced Normal mode frequencies of 4 horizontal springs, 3 masses

[Dixanadu]

Homework Statement

Hey guys.
The title says it all pretty much. We need to find the normal mode frequencies of a driven/forced system containing 3 equal masses connected by 4 springs of equal spring constant k.

Homework Equations

$F=m\ddot{x}$
Spring potential
$V = 0.5kx^{2}$
Force-potential relationship
$F = -\frac{dV}{dx}$
Natural frequency of the system
$ω_{0}^{2} = \frac{k}{m}$

The Attempt at a Solution

So, I'm gona type this up in word and show you guys what I've done. Obviously there are 3 masses, so we expect 3 normal mode frequencies. I've found two, I don't know how to get the third one.
https://imageshack.us/scaled/large/818/normalmodefrequencyques.jpg
Thanks again guys.

 

[TSny]

I haven't checked all the steps in detail. But, after you expanded your determinant, it looks like you have an overall factor of $(2\omega_0^2-\omega^2)$. You then divided out this overall factor. I think that's where you lost your 3rd solution.

 

[Dixanadu]

Yea, you're right...that's it. Thanks a lot man :)