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$a(η)[S/η^2 +Fη^2+Gη^4+Hη^6+J]+a'(η) [K/η+ηZ+η^3 C]+a''(η) [η^2 L+P]=0$

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.

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    If you multiply through by another $\eta^2$ it looks almost sensible, even exponents from 0 to 8 for $a,$ odd from 1 to 5 for $a',$ then again even from 2 to 4 for $a''.$ I would say it depends quite a bit on the $\pm$ signs for the coefficients and the relative sizes, finally how close $\eta$ gets to 0.2012-12-24

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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 $H$.

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    Thanks Mr Robert Israel, I aplied the Froebenius method again and find the equation's solution. Thank you very much.2012-12-26