If Mathematica refuses to solve this, it is likely that no exact solution exists. You might look for approximate solutions then, maybe expanding in power series (even though that wouldn't allow you to keep track of the delta term). Or you could generalize whatyou do in simple one-dimension potential scattering.
WKB method could help, too.
What have you tried? Do you have an expert you can ask at hand?
If you are a student, you can ask a professor at your university, otherwise you can try sending an email to someone who is expert in these sort of differential equations (maybe a professor of mathematical methods for physics, or of quantum mechanics).