I need to solve analytically a certain second order ODE, which takes the general form
y''(x) - p(z,x)*y(x) = 0.
where p(z,x) is a polynomial and y(x) is to be determined. The polynomial depends on a complex parameter z.
The problem is that I need a possibly approximate but analytical solution, perhaps in the form of some truncated series (but not the series in powers of x), not just numerical values of the solution.
Are there any techniques available? Some kind of successive approximations that really work (are convergent)? Someone has suggested to me the WKB approximation, but I think it is not suitable, because the above equation is not singularly perturbed.