I've put the ODE into a system.

We have:

which has Eigenvalues:

and Eigenvectors:

,

General Solution is

But this is where I am getting lost, I know that when the eigenvalues are complex I need to involve sin and cos, which is fine for a 1st order ODE but am getting a little stuck with this 2nd-order ODE.

Also, I need to find the solution in Real Form. i.e.

Thanks in advance to anyone who can help.