The solution to Stokes 5th involves an iterative solution, to solve equations 14 and 15 of Skjelbreia, L. & Hendrickson's paper. Your 1st guess (iteration starting value) can be obtained by assuming Airy theory.
(Note, I think the equations are 14 and 15, it's been a while since I've looked at this paper.)
Richard Ay
COADE, Inc.