Editing MathProjects/EigenvalueProblemN1 (section)

==Astrophysical Context==
A star that is composed entirely of a gas for which the pressure varies as the square of the gas density will have an equilibrium structure that is defined by, what astronomers refer to as, [[SSC/Structure/Polytropes#n_.3D_1_Polytrope|an <math>~n=1</math> polytrope]].  Inside such an equilibrium structure, the density of the gas will vary with radial position according to the expression,
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<math>~\frac{\rho}{\rho_c} = \frac{\sin x}{x} \, ,</math>
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where, <math>~\rho_c</math> is the density at the center of the star, and,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~x</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\pi\biggl(\frac{r}{R}\biggr) \, ,</math>
  </td>
</tr>
</table>
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where, <math>~R</math> is the radius of the equilibrium star.  Notice that, according to this expression, the density will drop to zero when <math>~r = R</math>, in which case, <math>~x = \pi</math>. If a star of this type is nudged out of equilibrium &#8212; for example, squeezed slightly &#8212; in such a way that it maintains its spherical symmetry, the star will begin to undergo periodic, radial oscillations about its original equilibrium radius.   The 2<sup>nd</sup>-order ODE whose solution is being sought in the above ''challenge'' is the equation that describes the behavior of these oscillations.  In particular, the function, 
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<math>~\mathcal{G}_\sigma(x) \equiv \frac{\delta x}{x}</math>
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describes the ''relative amplitude'' of the oscillation as a function of position, <math>~x</math>, within the star, and <math>~\sigma</math> gives the frequency of the oscillation.

<!--  COMMENT OUT (by mutual agreement) on 9/18/2015
==Suggested Eigenfunction by KV==

On 9/15/2015, KV recommended the following eigenfunction:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{G}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\exp\biggl[\tfrac{1}{4}\sigma^2 x^2 - \alpha \ln x\biggr]</math>
  </td>
</tr>
</table>
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where,
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<math>~\sigma^2 = \frac{1}{\pi^2} \biggl[ (1 - 8\alpha)^{1/2} + 2\alpha -1 \biggr]</math>
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END COMMENT  -->