Editing MathProjects/EigenvalueProblemN1 (section)

==The Challenge==

Formally, this is an eigenvalue problem.  

<table border="1" width="100%" cellpadding="10" align="center"><tr><td bgcolor="lightblue">
<table border="0" width="100%" cellpadding="5" align="left"><tr><td>

Find one or more analytic expression(s) for the (eigen)function, <math>~\mathcal{G}_\sigma(x)</math> &#8212; and, simultaneously, the unknown value of the (eigen)frequency, <math>~\sigma</math> &#8212; that satisfies the following 2<sup>nd</sup>-order, ordinary differential equation:

<div align="center">
<math>
(x^2\sin x ) \frac{d^2\mathcal{G}_\sigma}{dx^2} + 2 \biggl[ x \sin x +  x^2 \cos x \biggr]  \frac{d\mathcal{G}_\sigma}{dx} + 
\biggl[ \sigma^2 x^3  - 2\alpha ( \sin x - x\cos x ) \biggr]  \mathcal{G}_\sigma = 0 \, ,
</math><br />
</div>
<br />
where, <math>~\alpha</math> is a known constant.  The desired functional solution is subject to the following two boundary conditions:  <math>~\mathcal{G}_\sigma = 0</math> at <math>~x = 0</math>; and <math>~d\ln\mathcal{G}_\sigma/d\ln x = (\pi^2 \sigma^2/2 - \alpha)</math> at <math>~x = \pi</math>.  Note that, in the context of astrophysical discussions, the interval of <math>~x</math> that is of particular interest is <math>0 \le x \le \pi</math>.

</td></tr></table>
</td></tr>
</table>