Normal modes of a two mass, two spring system?

[Spiral1183]

Homework Statement

I have a system of two masses m₁ and m₂ coupled by two springs with constants k₁ and k₂. If m₁ and m₂ are equal what would be the normal modes for this system?

Homework Equations

Equations of motion for the system:
\begin{align*}
m_1\ddot{x}_1 &= -k_1x_1+k_2(x_2-x_1) \\
m_2\ddot{x}_2 &= -k_2(x_2-x_1)
\end{align*}

The Attempt at a Solution

I am having some serious trouble understanding how to come come up with the normal modes and thus the normal frequencies for this problem. I understand in a three spring system and have read examples of the two spring system, but am still struggling with how they came to their solution. Am I supposed to grind these problems out into a 4th order ODE, and if so where do I go from there?

 

[voko]

A normal mode is $x_1 = A \sin at, \ x_2 = B \sin at$. Substitute this and determine a, A, B.

 

[rahaverhma]

A normal mode is $x_1 = A \sin at, \ x_2 = B \sin at$. Substitute this and determine a, A, B.

Why same frequency ?