Eigenvectors of coupled 2 different mass system

[physics-1]

Homework Statement

Solve the coupled mass spring problem for two different masses. Similar to the Shankar example:
(d/dt)²x₁=-2*w₁*x₁ + w₁*x₂
(d/dt)²x₂=w₂*x₁ -2*w₂*x₂

where
w₁= k/m₁
w₂ = k/m₂

Homework Equations

Eigenvalue problem: UX=uX
Diagonalization: A=UDU^t
Exponential Matrices: e^A

The Attempt at a Solution

Starting off is simple enough. Taking the eigenvalue problem and solving for the eigenvalues.
L1 = -(w₁+w₂)+sqrt[ (w₁+w₂)² - 3w₁w₂]
and L2 is the same as L1 but a minus in front of the sqrt

My problem lies when plugging back in the eigenvalues to solve for the eigenvectors. It seems straightforward but it is not. When plugging eigenvalues back into the matrix, trying to solve for eigenvector solution for both equations within the matrix is a tricky task.
Does anyone here know any required tricks that are needed to solve for the eigenvectors?

Many books stray away from this problem and resort to the system where the two masses are the same.
Any help is really appreciated.

 

[physics-1]

Please disregard this posting. I have figured the solution.