Hi. This is what I think it is without looking so I'm not sure ok? You have second-order DEs which I can convert to first-order DEs just by adding more variables. I'd then have a system of first-order autonomous DEs which I can then in principle, compute the eigenvalues and vectors and then claim, the solution is:

where is the set of eigenvalues and , the set of eigenvectors.

I don't know. Is that right guys?