Editing MathProjects/EigenvalueProblemN1 (section)

==Context==

The challenge posed above is one of a set of closely related eigenvalue problems that arise in the context of the study of the pulsating stars and the governing  2<sup>nd</sup>-order ODE is often referred to as the ''Linear Adiabatic Wave Equation'' (LAWE).  In the most general context, the LAWE takes the form,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl[P \biggr]\frac{d^2\mathcal{G}_\sigma}{dx^2} + \biggl[\frac{4P}{x} 
+ P^' \biggr]\frac{d\mathcal{G}_\sigma}{dx} + \biggl[ \sigma^2 \rho + \frac{\alpha P^'}{x} \biggr]\mathcal{G}_\sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, ,</math>
  </td>
</tr>
</table>
</div>

where, <math>~P</math> and <math>~\rho</math> are both functions of <math>~x</math> that have different prescriptions for each specified astrophysics problem &#8212; see the table of examples presented below &#8212; and primes denote differentiation with respect to <math>~x</math>.  The symmetries associated with this broad set of eigenvalue problems can perhaps be better appreciated by rearranging terms in the LAWE to obtain, 

<div id="LAWE">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \sigma^2  \mathcal{G}_\sigma </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{P}{\rho} \biggl[ \frac{4\mathcal{G}_\sigma^'}{x}+ \mathcal{G}_\sigma^{' '} \biggr]  
+ \frac{P^'}{\rho} \biggl[ \frac{\alpha \mathcal{G}_\sigma}{x} + \mathcal{G}_\sigma^'\biggr] \, ,
</math>
  </td>
</tr>
</table>
or, equivalently,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \sigma^2  \rho \mathcal{G}_\sigma </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{P}{x^4} \frac{d}{dx}\biggl( x^4 \mathcal{G}_\sigma^' \biggr)  
+ \frac{P^'}{x^\alpha}\frac{d}{dx}\biggl(x^\alpha \mathcal{G}_\sigma\biggr) \, ,
</math>
  </td>
</tr>
</table>
or, equivalently,

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d}{dx}\biggl(x^4 P \mathcal{G}_\sigma^'\biggr) + \biggl[ \biggl( \sigma^2 + \frac{\alpha P^'}{x\rho} \biggr) x^4 \rho 
\biggr] \mathcal{G}_\sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 0 \, .</math>
  </td>
</tr>
</table>
</div>


<table border="1" cellpadding="5" align="center" width="90%">
<tr>
  <th align="center" colspan="4"><font size="+1">Properties of Analytically Defined Astrophysical Structures</font></th>
</tr>
<tr>
  <td align="center" width="10%">Model</td>
  <td align="center"><math>~\rho(x)</math>
  <td align="center"><math>~P(x)</math>
  <td align="center"><math>~P^'(x)</math>
</tr>
<tr>
  <td align="center">[[SSC/Stability/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Uniform-density]]</td>
  <td align="center"><math>~1</math>
  <td align="center"><math>~1 - x^2</math>
  <td align="center"><math>~-2x</math>
</tr>
<tr>
  <td align="center">[[SSC/Structure/OtherAnalyticModels#Linear_Density_Distribution|Linear]]</td>
  <td align="center"><math>~1-x</math>
  <td align="center"><math>~(1-x)^2(1 + 2x - \tfrac{9}{5}x^2)</math>
  <td align="center"><math>~-\tfrac{12}{5}x(1-x)(4-3x)</math>
</tr>
<tr>
  <td align="center">[[SSC/Structure/OtherAnalyticModels#Parabolic_Density_Distribution|Parabolic]]</td>
  <td align="center"><math>~1-x^2</math>
  <td align="center"><math>~(1-x^2)^2(1  - \tfrac{1}{2} x^2)</math>
  <td align="center"><math>~-x(1-x^2)(5-3x^2)</math>
</tr>
<tr>
  <td align="center">[[SSC/Stability/Polytropes#n_.3D_1_Polytrope|<math>~n=1</math> Polytrope]]</td>
  <td align="center"><math>~\frac{\sin }{ x}</math>
  <td align="center"><math>~\biggl[\frac{\sin x}{x}\biggr]^2</math>
  <td align="center"><math>~\frac{2}{x} \biggl[ \cos x - \frac{\sin x}{x} \biggr]
\frac{\sin x}{x}</math>
</tr>
</table>

Drawing the expressions for <math>~\rho(x)</math>, <math>~P(x)</math>, and <math>~P^'(x)</math> from the last row of this table and plugging them into this generic form of the LAWE leads to the ''specific'' statement of the astrophysically motivated eigenfunction problem presented above &#8212; inside the blue-framed box.  As is discussed in the [[#Analogous_Problem_with_Known_Analytic_Solutions|subsection that follows]], an analogous eigenvalue problem whose analytic solution is ''known'' comes from plugging expressions presented in the first row of this table into the generic form of the LAWE.