Your working seems fine all the way down to

Your integration of this DE might be correct, but that's the long way to go about it. You have to invert a rather messy function there, which might not be the easiest thing to do.

This is just a second-order ODE with constant coefficients, right? Why not just do guess-and-check? Surely, if you're doing a substitution this complicated, you know what the solution to the above DE is.