Can't you just solve this using inspired guesswork?
First evaluating the homogeneous solution, the characteristic equation is
Since these solutions are complex conjugates of the form , the solution is of the form . In this case and .
So the homogeneous solution is .
Now to look for a particular solution, since the RHS of your DE is , a particular solution might be .
So substituting into the DE gives
Equating like coefficients gives .
So your particular solution is .
Adding your homogeneous solution to your particular solution gives the general solution, so the general solution is