Like I said I made a mistake and . Previously, I assumed that the eigenvalues were purely imaginary which leads to a different solutions. This was incorrect. The form of the solution comes from Euler's Formula, i.e.,
.
So if our solution has complex eigenvalues the solution
where becomes
.
After further simplification the solution becomes
.
Do you see why my previous solution works if the eigenvalues are purely imaginary?
Again, I am sorry for the confusion.