I am looking for the analytic solution of this ODE (if it were one):(adsbygoogle = window.adsbygoogle || []).push({});

[tex]s^2G''+sG'-(1+s^2+s\text{coth}(s))G=-4s^2e^{-s}[/tex]

I have solved this equation numerically, it only gives one physically realizable configuration rejecting conveniently one of the homogenous solutions. I don't have those solutions BUT I have the asymptotic behavior of [tex]G(s)[/tex], which turns out to be [tex]G\sim As^{\sqrt{2}}[/tex] and [tex]G(s)\sim se^{-s}}[/tex] for small and large [tex]s[/tex] respectively, where A is a coefficient that I have worked out by means of a linear shooting.

When writting it on Maple in OdeAdvisor, it says to me that it is a Linear ODE (easy thing to know) and with Linear Symmetries. Does this last thing have something to do with Lie Symmetries?. May this ODE be solvable employing that theory, I don't have a clue.

Thanks.

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# Analytic solution of this, Advice.

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