> > > Hi, > > 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.
Depending on your p(z,x) expression and the boundary conditions of your problem, it might be possible to find an analytical solution to your problem. Do you have that information?
> 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?
Yes, a good number of them.
One possible approach is to employ any form of the Galerkin method. Basically, it essentially involves picking a function to approximate the exact solution and then adjusting it to the solution to your problem. Depending on the function you picked to approximate the solution, the Galerkin method can actually return it.
The finite element method can be seen as a very specific implementation of the Galerkin method. So, you can also explore that path.