How to Solve This Second-Order Non-Linear ODE Involving Functions a(r) and b(r)?

[ramparts]

I've run across a PDE that (since I've failed to take a PDE class!) I'm finding some difficulty in solving. Does anyone have any suggestions? It's on a function R(r,t), with functions a(r,t) and b(r,t) and a constant k. If it's easier to solve with a and b not having t-dependence (just being a(r) and b(r)) I'd be curious to know - I'm not sure if that affects things.

Here's the PDE:

R = k(-e^-2a R_tt + e^-2b R_rr +[a_r + b_r + 2/r] e^-2b R_r)

I'd really appreciate any help! I have a couple of ways of attacking this problem (I also have R defined in terms of derivatives of a and b) but I think a solution to this PDE in terms of a, b, r, and t would be incredibly useful. Thanks!

 

[ramparts]

Sorry - the first term (the derivatives in t) disappear. I don't expect any t dependence, so this is really just an ODE in r. Still, any advice would be appreciated :)