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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)

This equation comes from the eigenvalue problem of the graphene nanoring with spin orbit interaction and magnectic field using the mexican-hat potencial. To solve this equation I tried the froebenius method tha didn't work, and the maple software that didn't work too, the group have found a numerical solution using the runge-kunta method, but that's necessary an anathical or semi-analithical solution to compreend the real influence of spin-orbit interaction in graphene.

I would like to add that this is not a homework. In fact, this is an ongoing work with my advisor 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 this problem.

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Do you have any reasons to believe an analytical solution exists? General homogeneous second-order ODEs typically do not have one, and I don't expect one here unless the coefficients satisfy some special relationships. The ODE can be easily solved numerically, of course. – user7530 Dec 24 '12 at 22:40
Sounds like this would perhaps be better as a MathOverflow question. (Please make sure if you crosspost to there you include the link to this post there and vice versa.) – Alexander Gruber Dec 24 '12 at 22:40
may be you should change type of potential to get simpler equation. Also one should take into account order of constants. May be you can neglect some of them – Norbert Dec 24 '12 at 22:56
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. – Will Jagy Dec 24 '12 at 23:20

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. – Alexandre Cavalheiro Dias Dec 26 '12 at 9:57

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