I need to solve analytically a certain second order ODE having the form
u''(x) - p(x)*u(x) = 0
where p(x) is a known polynomial. The ODE most likely does not have a solution expressed by known analytical functions, so that by "analytical solution" I mean a solution that can be perhaps expressed by some series or maybe integral. I have tried a power series solution, but this approach is not useful for me for some reasons. Are there any other approaches that might be applicable? Some kind of successive approximations? Or other kinds of series solutions, not by power series? Please direct me to the relevant literature or web sites. I emphasise that I need to have a formula for the general solution (without taking into account boundary conditions), where all coefficients are given in an analytical form and can be subject to analytical processing. It's not sufficient for me just to compute the numerical values for a particular set of boundary conditions.