Your solution will be . You just set . You could very well have set this would just change the values of . The conventions is to set , but it doesn't matter.
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
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.