Question

Find the characteristic frequencies and the normal modes of the coupled harmonic oscillators:
m (x1)'' + k (x2) = 0
m (x2)'' + k (x1) = 0

Find also the solution that satisfies the initial conditions: x1(0) = 2; x_ 1(0) = 0; x2(0) = x_ 2(0) = 0.
**I figured out the first part, and have answers of (w1)^2 = k/m and (w2)^2 = -k/m and have also figured out the eigenvectors. However I am having problems formulating the solutions to the equations and solving for the coefficients.

Answer 1

So here are the eigenvectors:

For \[\omega _{1}^{2} = k/m = \left(\begin{matrix}1 \\ 1\end{matrix}\right)\] and \[\omega _{2}^{2} = -k/m = \left(\begin{matrix}1 \\ -1\end{matrix}\right)\]

Answer 2

addicting the two equations you will have and making y=x2+x1, you will have
my''+ky=0. after solving y[t]. you can make in the first equation:
mx1'' +k(y-x1)=0 and solve the differencial equation in x1.
may help you