2nd Order diffn eqn using Euler's eqn

Need help to find the particular integral of a 2nd order diffn eqn using Euler's eqn ;

I know that there are other methods of solution. Just want to understand the principle (Euler's) that i'm trying to apply to my type of solution.

particular integral,

,

By Euler's eqn

This is where i'm uncertain,

because question has on RHS

Question:

1) Should the Real part be included seeing that the question has the imaginary part (sin4x) only?

next, Remember

- Replace D with coefficient of x in

My final answer wasn't correct. Seek help using my method and not another approach.

Thanks.