TITLE: Method to find an analytical or semi-analytical solution for this diferential equation
QUESTION [7 upvotes]: $a(η)[S/η^2 +Fη^2+Gη^4+Hη^6+J]+a'(η) [K/η+ηZ+η^3 C]+a''(η) [η^2 L+P]=0$
where S,F,G,H,J,K,Z,C,L and P are constants
and a(η) is the function that's being sought.
This equation comes from the eigenvalue problem of the  graphene nano-ring with spin-orbit interaction and magnectic field using the mexican-hat potential. To solve this equation I tried the Froebenius method (it didn't work), and the Maple software (it didn't work either). The group has found a numerical solution using the Runge-Kunta method, but it's necessary to have an analytical or semi-analytical solution to comprehend the real influence of spin-orbit interaction in graphene.
I would like to add that this is not homework. In fact, this is an ongoing work with my adviser and after more than one month trying to obtain this solution I decided that I should ask for some help. I appreciate any reference or some hint that could help me with solving this problem.

REPLY [2 votes]: There's not much chance of a closed-form solution in general.  But you can get series solutions by Frobenius's method, since $\eta=0$ is a regular singular point.  The indicial equation is $r^2 + \dfrac{K-P}{P} r + \dfrac{S}{P} = 0$.
If $H=0$ Maple does come up with rather complicated closed-form solutions involving the HeunC function.  Based on this, for $H \ne 0$ you might be able to get solutions as series in powers of $H$.