Editing SSC/Structure/OtherAnalyticRamblings (section)

==Exploration2==

Let's begin with the LAWE written in the following form (see, for example, the [[MathProjects/EigenvalueProblemN1#Context|related context discussion]]):
<div align="center">
<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>~ \biggl(\frac{P}{\rho }\biggr)\frac{1}{x^4} \frac{d}{dx}\biggl( x^4 \mathcal{G}_\sigma^' \biggr)  
+ \biggl(\frac{P^'}{\rho }\biggr) \frac{1}{x^\alpha}\frac{d}{dx}\biggl(x^\alpha \mathcal{G}_\sigma\biggr) \, .
</math>
  </td>
</tr>
</table>
</div>

One advantage of beginning with this construction is that &#8212; as the following table shows &#8212; it might be reasonable to expect in general that both pre-factors &#8212; <math>~(P/\rho )</math>  and <math>~(P^'/\rho )</math> &#8212; will have a relatively simple mathematical form.  In addition, however, it appears as though both terms on the RHS ''want'' to be logarithmic derivatives.

<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>~\biggl[\frac{P(x)}{\rho(x)}\biggr]</math>
  <td align="center"><math>~\biggl[ \frac{P^'(x)}{\rho(x)} \biggr]</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)(1 + 2x - \tfrac{9}{5}x^2)</math>
  <td align="center"><math>~-\tfrac{12}{5}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)(1  - \tfrac{1}{2} x^2)</math>
  <td align="center"><math>~-x (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>~\frac{\sin x}{x}</math>
  <td align="center"><math>~\frac{2}{x} \biggl[ \cos x - \frac{\sin x}{x} \biggr] </math>
</tr>
</table>

===New Form of LAWE===

Multiplying the LAWE through by <math>~[\rho/(P\mathcal{G}_\sigma)]</math> gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- \sigma^2  \biggl(\frac{\rho}{P}\biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{\mathcal{G}_\sigma^'}{\mathcal{G}_\sigma }\biggr)\frac{1}{(x^4 \mathcal{G}_\sigma^')} \frac{d}{dx}\biggl( x^4 \mathcal{G}_\sigma^' \biggr)  
+ \biggl(\frac{P^'}{P}\biggr) \frac{1}{(x^\alpha \mathcal{G}_\sigma)}\frac{d}{dx}\biggl(x^\alpha \mathcal{G}_\sigma\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{d\ln \mathcal{G}_\sigma}{dx } \cdot \frac{d\ln( x^4 \mathcal{G}_\sigma^' )}{dx}
+ \frac{d\ln P}{dx } \cdot \frac{d\ln( x^\alpha \mathcal{G}_\sigma )}{dx}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{d\ln \mathcal{G}_\sigma}{dx } \biggl[ \frac{d\ln( x^4  )}{dx} + \frac{d\ln(  \mathcal{G}_\sigma^' )}{dx} \biggr]
+ \frac{d\ln P}{dx } \biggl[ \frac{d\ln( x^\alpha  )}{dx} + \frac{d\ln(  \mathcal{G}_\sigma )}{dx} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{d\ln \mathcal{G}_\sigma}{dx } \biggl[ \frac{d\ln( x^4  )}{dx} + \frac{d\ln(  \mathcal{G}_\sigma^' )}{dx} + \frac{d\ln P}{dx } \biggr]
+ \frac{d\ln P}{dx } \biggl[ \frac{d\ln( x^\alpha  )}{dx}  \biggr]
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~ - \frac{\mathcal{G}_\sigma}{P}\biggl[ \sigma^2  \rho + \frac{\alpha P^'}{x}  \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{1}{x^4 P } \biggl[ \frac{d( x^4 P  \mathcal{G}_\sigma^' )}{dx} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~~  \frac{d( x^4 P  \mathcal{G}_\sigma^' )}{dx} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \mathcal{G}_\sigma \biggl[ \sigma^2  x^4 \rho + \alpha x^3 P^'  \biggr] </math>
  </td>
</tr>

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

===Trial Logarithmic Eigenfunction===
Defining,
<div align="center">
<math>\mathcal{F}(x) \equiv \biggl[ \sigma^2   +  \frac{\alpha P^'}{x\rho}  \biggr] \, ,</math>
</div>

the LAWE becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \frac{d( x^4 P  \mathcal{G}_\sigma^' )}{dx} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-x^4 \rho \mathcal{F} \mathcal{G}_\sigma \, .</math>
  </td>
</tr>
</table>
</div>

====First Try====
Let's try an eigenvector of the form,
<div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{G}_\sigma </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~A(x) \ln P + B(x) \, ,</math>
  </td>
</tr>
</table>
in which case,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{G}_\sigma^' </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~A^' \ln P + \frac{A\cdot P^'}{P}+ B^' </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ x^4 P \mathcal{G}_\sigma^' </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x^4\biggl[ PA^' \ln P + A\cdot P^'+ PB^' \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~\mathrm{LHS} ~~\equiv \frac{d}{dx}\biggl[x^4 P \mathcal{G}_\sigma^' \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x^4\biggl[ P'A^' \ln P + PA^{' '} \ln P + PA^' \biggl(\frac{P^'}{P}\biggr)         
+ A'\cdot P^'+ PB^{' '} + A\cdot P^{' '}+ P'B^' \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 4x^3\biggl[ PA^' \ln P + A\cdot P^'+ PB^' \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x^4 \biggl\{\ln P \biggl[P'A^' + PA^{' '} + \frac{4}{x} \cdot PA^'
\biggr] + 
\biggl[  A^' P^' + A'\cdot P^'+ PB^{' '} + A\cdot P^{' '}+ P'B^' + \frac{4}{x} \biggl( A\cdot P^'+ PB^'  \biggr) \biggr]\biggr\} \, .
</math>
  </td>
</tr>
</table>

</div>
Now, in order for this expression to match the RHS of the LAWE, we must have, first of all,
<div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{F} A</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\frac{1}{\rho}\biggl[ P'A^' + PA^{' '} + \frac{4}{x} \cdot PA^'\biggr] \, ;</math>
  </td>
</tr>
</table>
and, second,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathcal{F} B</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\frac{1}{\rho}\biggl[A^' P^' + A'\cdot P^'+ PB^{' '} + A\cdot P^{' '}+ P'B^' + \frac{4}{x} \biggl( A\cdot P^'+ PB^'  \biggr) \biggr] \, .</math>
  </td>
</tr>
</table>

</div>

<font color="red">'''Case 1''' (Parabolic)</font>:
<div align="center">
<math>\mathcal{F}(x) \equiv \biggl[ \sigma^2   - \alpha(5 - 3x^2) \biggr] = f_0 + f_2 x^2 \, ,</math>
</div>
where,
<div align="center">
<math>f_0 \equiv \sigma^2 - 5\alpha</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>f_2 \equiv 3\alpha \, .</math>
</div>
Also, the first condition is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- 2( f_0 + f_2 x^2 )A</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2( -5 + 3x^2 ) xA^' + (2-3x^2 + x^4) \biggl[A^{' '} + \frac{4A^'}{x}  \biggr] \, .</math>
  </td>
</tr>
</table>

So, if we adopt a polynomial expression for the function, <math>~A(x)</math>, of the form,
<div align="center">
<math>~A(x) = a_0 + a_2 x^2 + a_4 x^4 \, ,</math>
</div>


<div align="center">
<math>~\Rightarrow ~~~~ A^' = 2a_2 x + 4a_4 x^3</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp; 
<math>A^{' '} = 2a_2 + 12a_4 x^2 \, ,</math>
</div>
the condition becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~- 2( f_0 + f_2 x^2 )(a_0 + a_2 x^2 + a_4 x^4)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~( -10 + 6x^2 ) (2a_2 x^2 + 4a_4 x^4) + (2-3x^2 + x^4) \biggl[(2a_2 + 12a_4 x^2) + 4 (2a_2  + 4a_4 x^2) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ ( f_0 + f_2 x^2 )(a_0 + a_2 x^2 + a_4 x^4)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~( 10 - 6x^2 ) (a_2 x^2 + 2 a_4 x^4) - (2-3x^2 + x^4) (5a_2 + 14a_4 x^2)  </math>
  </td>
</tr>
</table>

So, the coefficients of each even power of <math>~\chi_0^n</math> are:
<div align="center" id="FirstTable">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="right"><math>~\chi_0^0</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~f_0 a_0 + 10 a_2</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~f_0  a_2 + f_2 a_0 -10a_2 + 28a^4 - 15a_2</math>
 </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~f_0 a_4 + f_2 a_2 -20a_4 +6a_2 -42a_4 + 5a_2</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^6</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~f_2 a_4 +12a_4 +14a_4</math>
  </td>
</tr>
</table>
</div>
This does not seem to work.

===Another Trial===

Start with a form of the LAWE found midway through the [[SSC/Structure/OtherAnalyticRamblings#New_Form_of_LAWE|above derivation]], namely,

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

<tr>
  <td align="right">
<math>~- \biggl[ \sigma^2  \biggl(\frac{\rho}{P}\biggr) + \frac{d\ln P}{dx } \cdot \frac{d\ln( x^\alpha  )}{dx}  \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{d\ln \mathcal{G}_\sigma}{dx } \biggl[ \frac{d\ln( x^4 P  \mathcal{G}_\sigma^' )}{dx} \biggr] \, ;
</math>
  </td>
</tr>
</table>
</div>
and multiplying through by <math>~x^2</math> gives,

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

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ -\frac{d\ln P}{d\ln x }  \biggl[ \sigma^2  \biggl(\frac{P^'}{x \rho}\biggr)^{-1} + \alpha \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{d\ln \mathcal{G}_\sigma}{d\ln x } \biggl[ \frac{d\ln( \mathcal{G}_\sigma^' )}{d\ln x} + \frac{d\ln( P )}{d\ln x} 
+ \frac{d\ln( x^4  )}{d\ln x} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ -\frac{d\ln P}{d\ln x }  \biggl[\alpha +  \sigma^2  \biggl(\frac{P^'}{x \rho}\biggr)^{-1} + \frac{d\ln \mathcal{G}_\sigma}{d\ln x } \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{d\ln \mathcal{G}_\sigma}{d\ln x } \biggl[ \frac{d\ln( \mathcal{G}_\sigma^' )}{d\ln x} +4 \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\frac{d\ln \mathcal{G}_\sigma}{d\ln x } </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{d\ln \mathcal{G}_\sigma}{d\ln x } \cdot \frac{d\ln( \mathcal{G}_\sigma^' )}{d\ln x}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{x^2 \mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma} \, .</math>
  </td>
</tr>
</table>
</div>

====Consider Parabolic Case====
In the case of a parabolic density distribution, the LAWE becomes,

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

<tr>
  <td align="right">
<math>~\frac{2x^2(5-3x^2)}{(1-x^2)(2-x^2)}  \biggl[\alpha -  \sigma^2  \biggl(5-3x^2\biggr)^{-1} + \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma} \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{x^2 \mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}\biggr) +4 \cdot \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \frac{2}{(1-x^2)(2-x^2)}  \biggl[ \biggl( \alpha + \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}\biggr)(5-3x^2) -\sigma^2 \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}\biggr) +\frac{4}{x^2} \cdot \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}  
</math>
  </td>
</tr>
</table>
</div>


Let's try,
<div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center">
<math>~\mathcal{G}_\sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^m \, ,</math>
  </td>
</tr>
</table>
which implies,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center">
<math>~\mathcal{G}_\sigma^'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~n(a_0 + a_2x^2)^{n-1}(2a_2x) \cdot (b_0 + b_2x^2)^m +m (a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^{m-1}(2b_2x)</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~\Rightarrow ~~~~ \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~n(a_0 + a_2x^2)^{-1}(2a_2x^2)  +m  (b_0 + b_2x^2)^{-1}(2b_2x^2) </math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2x^2}{(a_0 + a_2x^2) (b_0 + b_2x^2)} \biggl[ n a_2 (b_0 + b_2x^2)  +mb_2  (a_0 + a_2x^2) \biggr] </math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2x^2}{(a_0 + a_2x^2) (b_0 + b_2x^2)} \biggl[ (n a_2 b_0 + mb_2 a_0)  +(na_2 b_2+ mb_2 a_2)x^2\biggr] \, ,</math>
  </td>
</tr>
</table>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="center">
<math>~\mathcal{G}_\sigma^{' '}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~n m (a_0 + a_2x^2)^{n-1}(2a_2x) \cdot (b_0 + b_2x^2)^{m-1}(2b_2x) + n(a_0 + a_2x^2)^{n-1}(2a_2) \cdot (b_0 + b_2x^2)^m  + n(n-1)(a_0 + a_2x^2)^{n-2}(2a_2x)^2 \cdot (b_0 + b_2x^2)^m </math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+m n(a_0 + a_2x^2)^{n-1}(2a_2x)  \cdot (b_0 + b_2x^2)^{m-1}(2b_2x) +m (a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^{m-1}(2b_2) 
+m(m-1) (a_0 + a_2x^2)^n \cdot (b_0 + b_2x^2)^{m-2}(2b_2x)^2</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~\Rightarrow ~~~~ \frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~8n m a_2b_2 x^2 (a_0 + a_2x^2)^{-1}\cdot (b_0 + b_2x^2)^{-1} + n2a_2 (a_0 + a_2x^2)^{-1}   + n(n-1)4a_2^2 x^2 (a_0 + a_2x^2)^{-2}  
+m2b_2 (b_0 + b_2x^2)^{-1} +m(m-1)4 b_2^2 x^2 (b_0 + b_2x^2)^{-2}</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2n a_2(b_0 + b_2x^2) + 2m b_2  (a_0 + a_2x^2)}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}  + \biggl[ \frac{4n(n-1) a_2^2 }{ (a_0 + a_2x^2)^{2}}  
+ \frac{8n m a_2b_2}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}+ \frac{4m(m-1) b_2^2 }{(b_0 + b_2x^2)^{2}} \biggr]x^2
</math>
  </td>
</tr>
</table>
</div>

So, we have for the LAWE:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
LHS
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{2}{(1-x^2)(2-x^2)}  \biggl[ \biggl( \alpha + \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma}\biggr)(5-3x^2) -\sigma^2 \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \frac{2}{(1-x^2)(2-x^2)(a_0 + a_2x^2) (b_0 + b_2x^2)}  
\biggl\{ \biggl[ \alpha(a_0 + a_2x^2) (b_0 + b_2x^2) + 2x^2(n a_2 b_0 + mb_2 a_0)  + 2x^4 (na_2 b_2+ mb_2 a_2) \biggr](5-3x^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
-\sigma^2 (a_0 + a_2x^2) (b_0 + b_2x^2) \biggr\} \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
RHS
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{\mathcal{G}_\sigma^{' '}}{\mathcal{G}_\sigma}\biggr) +\frac{4}{x^2} \cdot \frac{x \mathcal{G}_\sigma^'}{\mathcal{G}_\sigma} </math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2n a_2(b_0 + b_2x^2) + 2m b_2  (a_0 + a_2x^2)}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}  + \biggl[ \frac{4n(n-1) a_2^2 }{ (a_0 + a_2x^2)^{2}}  
+ \frac{8n m a_2b_2}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}+ \frac{4m(m-1) b_2^2 }{(b_0 + b_2x^2)^{2}} \biggr]x^2
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{8}{(a_0 + a_2x^2) (b_0 + b_2x^2)} \biggl[ (n a_2 b_0 + mb_2 a_0)  +(na_2 b_2+ mb_2 a_2)x^2\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(a_0 + a_2x^2)(b_0 + b_2x^2)} \biggl\{ 2n a_2(b_0 + b_2x^2) + 2m b_2  (a_0 + a_2x^2)  
+ 8(n a_2 b_0 + mb_2 a_0)  + 8(na_2 b_2+ mb_2 a_2)x^2
</math>
  </td>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[8n m a_2b_2+ \frac{4n(n-1) a_2^2(b_0 + b_2x^2) }{ (a_0 + a_2x^2)}  
+ \frac{4m(m-1) b_2^2(a_0 + a_2x^2) }{(b_0 + b_2x^2)} \biggr]x^2
\biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

Putting these together gives,

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

<tr>
  <td align="right">
<math>~ 0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \alpha(a_0 + a_2x^2) (b_0 + b_2x^2) + 2x^2(n a_2 b_0 + mb_2 a_0)  + 2x^4 (na_2 b_2+ mb_2 a_2) \biggr](5-3x^2) 
-\sigma^2 (a_0 + a_2x^2) (b_0 + b_2x^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[ n a_2(b_0 + b_2x^2) + m b_2  (a_0 + a_2x^2)  + 4(n a_2 b_0 + mb_2 a_0)  + 4(na_2 b_2+ mb_2 a_2)x^2+ 4n m a_2b_2x^2
\biggr](1-x^2)(2-x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{(1-x^2)(2-x^2)}{ (a_0 + a_2x^2)(b_0 + b_2x^2)}\biggl[2n(n-1) a_2^2(b_0 + b_2x^2)^2   + 2m(m-1) b_2^2(a_0 + a_2x^2)^2 \biggr]x^2 \, .
</math>
  </td>
</tr>
</table>
</div>

----
=====First Guess=====
Now, if we are very lucky, we will find that,
<div align="center">
<math>~(a_0 + a_2x^2) = (1-x^2)</math> &nbsp; &nbsp;&nbsp; <math>~\Rightarrow</math>
&nbsp; &nbsp;&nbsp; <math>~a_0 = 1</math> &nbsp; &nbsp;&nbsp; and
&nbsp; &nbsp;&nbsp; <math>~a_2 = -1</math>;
</div>
and, simultaneously,
<div align="center">
<math>~(b_0 + b_2x^2) = (2-x^2)</math> &nbsp; &nbsp;&nbsp; <math>~\Rightarrow</math>
&nbsp; &nbsp;&nbsp; <math>~b_0 = 2</math> &nbsp; &nbsp;&nbsp; and
&nbsp; &nbsp;&nbsp; <math>~b_2 = -1</math>.
</div>
In this case, the fractional coefficient in the last term of the LAWE will become unity and the LAWE becomes,

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

<tr>
  <td align="right">
<math>~ 0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \alpha(1-x^2) (2-x^2) - 2x^2(2n  + m)  + 2x^4 (n+ m) \biggr](5-3x^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[\sigma^2 - n (2-x^2) - m  (1-x^2)  - 4(2n  + m)  + 4(n + m )x^2+ 4n m x^2 \biggr](1-x^2)(2-x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[2n(n-1) (2-x^2)^2   + 2m(m-1) (1-x^2)^2 \biggr]x^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 2\alpha - x^2(3\alpha +4n  + 2m)  + x^4 (\alpha + 2n+ 2m) \biggr](5-3x^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[(-\sigma^2 + 10n + 5m  )  - x^2( 5n+5m+4nm  )\biggr] (2-3x^2+x^4)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl\{ [8n(n-1) +2m(m-1) ] - [8n(n-1) + 4m(m-1)  ] x^2 + [ 2n(n-1)+ 2m(m-1) ] x^4 \biggr\} x^2 \, .
</math>
  </td>
</tr>
</table>
</div>

So, the coefficients of each even power of <math>~\chi_0^n</math> are:
<div align="center" id="FirstTable">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="right"><math>~\chi_0^0</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~10\alpha + 2(-\sigma^2 + 10n + 5m  )</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~-6\alpha - 5(3\alpha +4n  + 2m) -2( 5n+5m+4nm  ) - 3(-\sigma^2 + 10n + 5m  ) - [8n(n-1) +2m(m-1) ]</math>
 </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~5(\alpha + 2n+ 2m)+3(3\alpha +4n  + 2m) + (-\sigma^2 + 10n + 5m  )  +3( 5n+5m+4nm  ) + [8n(n-1) + 4m(m-1)  ] </math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^6</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~-3(\alpha + 2n+ 2m) - ( 5n+5m+4nm  ) - [ 2n(n-1)+ 2m(m-1) ] </math>
  </td>
</tr>
</table>
</div>


Now let's begin simplification.  The <math>~x^0</math> coefficient implies,
<div align="center">
<math>~(-\sigma^2 + 10n + 5m  )=-5\alpha \, .</math>
</div>

Hence,
<div align="center">
<table border="1" cellpadding="8" align="center">

<tr>
  <td align="right"><math>~\chi_0^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~-6\alpha - 5( 4n  + 2m) -2( 5n+5m+4nm  )  - [8n(n-1) +2m(m-1) ]</math>
 </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~9\alpha + 5(2n+ 2m)+3(4n  + 2m)  +3( 5n+5m+4nm  ) + [8n(n-1) + 4m(m-1)  ] </math>
  </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^6</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~-3\alpha -3(2n+ 2m) - ( 5n+5m+4nm  ) - [ 2n(n-1)+ 2m(m-1) ] </math>
  </td>
</tr>
</table>
</div>

Using the <math>~x^6</math> coefficient to define <math>~\alpha</math>, that is, setting,
<div align="center">
<math>~3\alpha = \{-3(2n+ 2m) - ( 5n+5m+4nm  ) - [ 2n(n-1)+ 2m(m-1) ]\} \, ,</math>
</div>

means that the other two coefficient expressions are,
<div align="center">
<table border="1" cellpadding="8" align="center">

<tr>
  <td align="right"><math>~\chi_0^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~6(2n+ 2m) + 2 ( 5n+5m+4nm  ) + 4 [ n(n-1)+ m(m-1) ] - 5( 4n  + 2m) -2( 5n+5m+4nm  )  - [8n(n-1) +2m(m-1) ]</math>
 </td>
</tr>

<tr>
  <td align="right"><math>~\chi_0^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~-9(2n+ 2m) - 3( 5n+5m+4nm  ) - 6[ n(n-1)+ m(m-1) ] + 5(2n+ 2m)+3(4n  + 2m)  +3( 5n+5m+4nm  ) + [8n(n-1) + 4m(m-1)  ] </math>
  </td>
</tr>
</table>
</div>

The only question remaining is, what pair of values for <math>~(n, m)</math> result in both of these expressions going to zero?  Simplifying the first expression gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~6(2n+ 2m) + 2 ( 5n+5m+4nm  ) + 4 [ n(n-1)+ m(m-1) ] - 5( 4n  + 2m) -2( 5n+5m+4nm  )  - [8n(n-1) +2m(m-1) ]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2m  -8n -4n(n-1) + 2m(m-1) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 2[m^2 - 2n(n+1)] \, . </math>
  </td>
</tr>
</table>
</div>

Simplifying the second expression gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-9(2n+ 2m) - 3( 5n+5m+4nm  ) - 6[ n(n-1)+ m(m-1) ] + 5(2n+ 2m)+3(4n  + 2m)  +3( 5n+5m+4nm  ) + [8n(n-1) + 4m(m-1)  ] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4n - 2m   + 2n(n-1) - 2m(m-1) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2[n(n+1) -m^2] </math>
  </td>
</tr>
</table>
</div>

Okay.  Because it is not possible for both of these last two constraints to be simultaneously satisfied, I conclude that this last, specific eigenfunction guess is incorrect.  

----
=====Second Guess=====
Let's try again, keeping the same values of the <math>~b_0</math> and <math>~b_2</math> &#8212; that is,
<div align="center">
<math>~(b_0 + b_2x^2) = (2-x^2)</math> &nbsp; &nbsp;&nbsp; <math>~\Rightarrow</math>
&nbsp; &nbsp;&nbsp; <math>~b_0 = 2</math> &nbsp; &nbsp;&nbsp; and
&nbsp; &nbsp;&nbsp; <math>~b_2 = -1</math>
</div>

&#8212; but leaving the values of <math>~a_0</math> and <math>~a_2</math> unspecified.  In this case, the LAWE becomes,

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

<tr>
  <td align="right">
<math>~ 0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \alpha(a_0 + a_2x^2) (2 - x^2) + 2x^2(2n a_2 - m a_0)  - 2x^4 (na_2 + m a_2) \biggr](5-3x^2)(a_0 + a_2x^2) 
-\sigma^2 (a_0 + a_2x^2)^2 (2 - x^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[ n a_2(2 - x^2) -m  (a_0 + a_2x^2)  + 4(2n a_2 - m a_0)  - 4(na_2 + m a_2)x^2 - 4n m a_2 x^2
\biggr](1-x^2)(2-x^2)(a_0 + a_2x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl[2n(n-1) a_2^2(2 - x^2)^2   + 2m(m-1) (a_0 + a_2x^2)^2 \biggr]x^2 (1-x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \alpha [2a_0 ] + x^2[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0)  ]  - x^4 [(2na_2 + 2ma_2 ) + a_2\alpha  ]\biggr\} [ 5a_0 + (5a_2-3a_0)x^2 -3a_2x^4] 
-\sigma^2 [ 2a_0 + (2a_2-a_0)x^2 - a_2x^4 ] (a_0+a_2x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[ ( 5m a_0 - 10n a_2)  +  (4n m a_2 + 5na_2 + 5m a_2)x^2 \biggr] (1-x^2)[ 2a_0 + (2a_2-a_0)x^2 - a_2x^4 ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl\{ [ 8n(n-1) a_2^2 + 2m(m-1)a_0^2 ] + [ -8n(n-1) a_2^2 + 4m(m-1)a_0 a_2 ]x^2 + [ 2n(n-1) a_2^2  + 2m(m-1)a_2^2 ]x^4    \biggr\} x^2 (1-x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \alpha [2a_0 ] + x^2[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0)  ]  - x^4 [(2na_2 + 2ma_2 ) + a_2\alpha  ]\biggr\} [ 5a_0 + (5a_2-3a_0)x^2 -3a_2x^4] 
</math>
  </td> 
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-\sigma^2\biggl\{ 2a_0^2 + [2a_0a_2 + a_0(2a_2-a_0)]x^2 +[a_2(2a_2-a_0) -a_0a_2]x^4 - a_2^2 x^6  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl\{ [ ( 5m a_0 - 10n a_2)  ] + [(4n m a_2 + 5na_2 + 5m a_2)-  ( 5m a_0 - 10n a_2) ]x^2 - [ 4n m a_2 + 5na_2 + 5m a_2  ]x^4 \biggr\}
[ 2a_0 + (2a_2-a_0)x^2 - a_2x^4 ]
</math>
  </td> 
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \biggl\{ [ 8n(n-1) a_2^2 + 2m(m-1)a_0^2 ]x^2 + [ -8n(n-1) a_2^2 + 4m(m-1)a_0 a_2 ]x^4 + [ 2n(n-1) a_2^2  + 2m(m-1)a_2^2 ]x^6    \biggr\} (1-x^2) \, .
</math>
  </td> 
</tr>

</table>
</div>


So, the coefficients of each even power of <math>~x^n</math> are:
<div align="center" id="FirstTable">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="right"><math>~x^0</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~
10a_0^2 \alpha - 2a_0^2\sigma^2 + 2a_0[ ( 5m a_0 - 10n a_2)  ]
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~5a_0[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0)  ] + \alpha [2a_0 ](5a_2-3a_0)-\sigma^2[2a_0a_2 + a_0(2a_2-a_0)]
</math><p>
<math>~+ 2a_0[(4n m a_2 + 5na_2 + 5m a_2)-  ( 5m a_0 - 10n a_2) ] + (2a_2-a_0)[ ( 5m a_0 - 10n a_2)  ] - [ 8n(n-1) a_2^2 + 2m(m-1)a_0^2 ]
</math></p>
 </td>
</tr>

<tr>
  <td align="right"><math>~x^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ - 5a_0[(2na_2 + 2ma_2 ) + a_2\alpha  ] + (5a_2-3a_0)[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0)  ] - 6a_0 a_2\alpha -\sigma^2[a_2(2a_2-a_0) -a_0a_2] 
</math><p>
<math>~- 2a_0[ 4n m a_2 + 5na_2 + 5m a_2  ] + (2a_2-a_0)[(4n m a_2 + 5na_2 + 5m a_2)-  ( 5m a_0 - 10n a_2) ] - a_2(5ma_0 - 10na_2)
</math></p><p>
<math>~-[ -8n(n-1) a_2^2 + 4m(m-1)a_0 a_2 ] + [ 8n(n-1) a_2^2 + 2m(m-1)a_0^2 ]
</math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^6</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ -3a_2[(4n a_2 - 2m a_0) + \alpha (2a_2-a_0)  ] - (5a_2-3a_0)[(2na_2 + 2ma_2 ) + a_2\alpha  ] +\sigma^2 a_2^2
</math><p>
<math>~- (2a_2-a_0)[ 4n m a_2 + 5na_2 + 5m a_2  ] -  a_2[(4n m a_2 + 5na_2 + 5m a_2)-  ( 5m a_0 - 10n a_2) ] 
</math></p><p>
<math>~- [ 2n(n-1) a_2^2  + 2m(m-1)a_2^2 ] + [ -8n(n-1) a_2^2 + 4m(m-1)a_0 a_2 ]
</math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^8</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ 3a_2 a_2\alpha + a_2[ 4n m a_2 + 11na_2 + 11m a_2  ] + [ 2n(n-1) a_2^2  + 2m(m-1)a_2^2 ]
</math>
  </td>
</tr>
</table>
</div>


After simplification:
<div align="center" id="FirstTable">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="right"><math>~x^0</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~
10a_0^2 \alpha - 2a_0^2\sigma^2 + 10m a_0^2 - 20n a_0a_2
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~\alpha (20a_0a_2-11a_0^2)    -\sigma^2[4a_0a_2 -a_0^2]  
</math><p>
<math>~+ 60na_0a_2 -20na_2^2  + 20m a_0a_2 -25m a_0^2   + 8n m a_0a_2 
- [ 8n(n-1) a_2^2 + 2m(m-1)a_0^2 ]
</math></p>
 </td>
</tr>

<tr>
  <td align="right"><math>~x^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ \alpha (10a_2^2 - 22a_0a_2+3a_0^2) -\sigma^2 (2a_2^2 -2a_0a_2)
-47n a_0a_2  +  60n a_2^2 - 50ma_0a_2 +  11m a_0^2 + 10m a_2^2-12n m a_0a_2  + 8n m a_2^2
</math><p>
<math>~+ 16n(n-1) a_2^2 - 4m(m-1)a_0 a_2  + 2m(m-1)a_0^2 
</math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^6</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ \alpha (-11a_2^2 + 6a_0 a_2) +\sigma^2 a_2^2 -47n a_2^2 + 11na_0a_2+ 22 m a_0a_2 -25ma_2^2   -12n m a_2^2  + 4n m a_0a_2
</math><p>
<math>~-10n(n-1) a_2^2  - 2m(m-1)a_2^2  + 4m(m-1)a_0 a_2 
</math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^8</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ \{ 3\alpha + [ 4n m  + 11n + 11m   ] + [ 2n(n-1)   + 2m(m-1) ]\}a_2^2
</math>
  </td>
</tr>
</table>
</div>



----

=====Third Guess=====

Let's try again, keeping the same values of the <math>~b_0</math> and <math>~b_2</math> &#8212; that is,
<div align="center">
<math>~(b_0 + b_2x^2) = (2-x^2)</math> &nbsp; &nbsp;&nbsp; <math>~\Rightarrow</math>
&nbsp; &nbsp;&nbsp; <math>~b_0 = 2</math> &nbsp; &nbsp;&nbsp; and
&nbsp; &nbsp;&nbsp; <math>~b_2 = -1</math>
</div>

&#8212; but leaving the values of <math>~a_0</math> and <math>~a_2</math> unspecified.  In this case, the LAWE becomes,

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

<tr>
  <td align="right">
<math>~ 0
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \alpha(a_0 + a_2x^2) (2 - x^2) + 2x^2(2n a_2 - m a_0)  - 2a_2 x^4 (n + m ) \biggr](5-3x^2) 
-\sigma^2 (a_0 + a_2x^2) (2 - x^2) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[-  n a_2(2 - x^2) + m  (a_0 + a_2x^2)  - 4(n a_2 2 - m a_0)  + 4(na_2 + m a_2)x^2 + 4n m a_2 x^2
\biggr](1-x^2)(2-x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{1}{ (a_0 + a_2x^2)}\biggl[2n(n-1) a_2^2(2 - x^2)^2   + 2m(m-1) (a_0 + a_2x^2)^2 \biggr](1-x^2)x^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{[2a_0\alpha   ] + [\alpha(- a_0 +2a_2) + (4n a_2 - 2m a_0)  ]x^2 + [-a_2\alpha  - 2a_2 (n + m )  ]x^4  \biggr\} (5-3x^2) 
-\sigma^2 [2a_0 + (-a_0 + 2a_2)x^2 -a_2x^4]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \biggl[( -10na_2 + 5ma_0 )  + (5ma_2 + 5na_2  + 4n m a_2 )x^2 \biggr](2-3x^2 +x^4)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{1}{ (a_0 + a_2x^2)}\{ [ 8n(n-1)a_2^2 +2m(m-1)a_0^2 ]  + [ -8n(n-1)a_2^2 +4m(m-1)a_0 a_2 ]x^2 + [ 2n(n-1)a_2^2 +2m(m-1)a_2^2 ]x^4  \} (x^2-x^4) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ [10a_0\alpha - 2a_0\sigma^2  ] + [5\alpha(- a_0 +2a_2) + 5(4n a_2 - 2m a_0) -6a_0\alpha  + (a_0 - 2a_2)\sigma^2  ]x^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ [-5a_2\alpha  - 10a_2 (n + m ) -3\alpha(- a_0 +2a_2) -3 (4n a_2 - 2m a_0) + a_2\sigma^2]x^4  
+ [3a_2\alpha  +6a_2 (n + m )  ]x^6  \biggr\}  (a_0 + a_2x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+\biggl\{ [2( -10na_2 + 5ma_0 )]  + [2 (5ma_2 + 5na_2  + 4n m a_2 ) -3( -10na_2 + 5ma_0 )]x^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ [ -3 (5ma_2 + 5na_2  + 4n m a_2 ) + ( -10na_2 + 5ma_0 )]x^4  + (5ma_2 + 5na_2  + 4n m a_2 )x^6 
\biggr\} (a_0 + a_2x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \{ [ 8n(n-1)a_2^2 +2m(m-1)a_0^2 ]  + [ -8n(n-1)a_2^2 +4m(m-1)a_0 a_2 ]x^2 + [ 2n(n-1)a_2^2 +2m(m-1)a_2^2 ]x^4  \} (x^2-x^4) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{ [10a_0\alpha - 2a_0\sigma^2  ] + [\alpha(- 11a_0 +10a_2) + (20n a_2 - 10m a_0)   + (a_0 - 2a_2)\sigma^2  ]x^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ [\alpha(3a_0 - 11a_2) + (-22n a_2 +6m a_0 - 10a_2 m)  + a_2\sigma^2]x^4  
+ [3a_2\alpha  +6a_2 (n + m )  ]x^6  \biggr\}  (a_0 + a_2x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \biggl\{ [ -20na_2 + 10ma_0 ]  + [10ma_2 + 40na_2  + 8n m a_2   -15ma_0 ]x^2 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ [ -15ma_2 - 25na_2  -12 n m a_2  + 5ma_0 ]x^4  + [5ma_2 + 5na_2  + 4n m a_2 ]x^6 
\biggr\} (a_0 + a_2x^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \{ [ 8n(n-1)a_2^2 +2m(m-1)a_0^2 ]x^2  + [ -8n(n-1)a_2^2 +4m(m-1)a_0 a_2 ]x^4 + [ 2n(n-1)a_2^2 +2m(m-1)a_2^2 ]x^6  \} (1-x^2) 
</math>
  </td>
</tr>
</table>
</div>

So, the coefficients of each even power of <math>~x^n</math> are:
<div align="center" id="FirstTable">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="right"><math>~x^0</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~a_0[10a_0\alpha - 2a_0\sigma^2  ] + a_0 [ -20na_2 + 10ma_0 ]</math>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~a_0  [\alpha(- 11a_0 +10a_2) + (20n a_2 - 10m a_0)   + (a_0 - 2a_2)\sigma^2  ] + a_2[10a_0\alpha - 2a_0\sigma^2  ] 
+ a_0[10ma_2 + 40na_2  + 8n m a_2   -15ma_0 ] 
</math><p><math>
+ a_2[ -20na_2 + 10ma_0 ] - [ 8n(n-1)a_2^2 +2m(m-1)a_0^2 ]
</math></p>
 </td>
</tr>

<tr>
  <td align="right"><math>~x^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ 
a_0 [\alpha(3a_0 - 11a_2) + (-22n a_2 +6m a_0 - 10a_2 m)  + a_2\sigma^2] + a_2[\alpha(- 11a_0 +10a_2) + (20n a_2 - 10m a_0)   + (a_0 - 2a_2)\sigma^2  ]
</math><p>
<math>~
+ a_0 [ -15ma_2 - 25na_2  -12 n m a_2  + 5ma_0 ] + a_2[10ma_2 + 40na_2  + 8n m a_2   -15ma_0 ]
</math></p><p>
<math>
~ - [ -8n(n-1)a_2^2 +4m(m-1)a_0 a_2 ] + [ 8n(n-1)a_2^2 +2m(m-1)a_0^2 ]
</math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^6</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ 
a_0[3a_2\alpha  +6a_2 (n + m )  ]  + a_2[\alpha(3a_0 - 11a_2) + (-22n a_2 +6m a_0 - 10a_2 m)  + a_2\sigma^2]
</math><p>
<math>~
a_0[5ma_2 + 5na_2  + 4n m a_2 ]  + a_2[ -15ma_2 - 25na_2  -12 n m a_2  + 5ma_0 ]
</math></p><p>
<math>~
-[ 2n(n-1)a_2^2 +2m(m-1)a_2^2 ] + [ -8n(n-1)a_2^2 +4m(m-1)a_0 a_2 ]
</math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^8</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ 
a_2[3a_2\alpha  +6a_2 (n + m )  ] + a_2[5ma_2 + 5na_2  + 4n m a_2 ] + [ 2n(n-1)a_2^2 +2m(m-1)a_2^2 ] 
</math>
  </td>
</tr>

</table>
</div>


After simplification:
<div align="center" id="FirstTable">
<table border="1" cellpadding="8" align="center">
<tr>
  <td align="right"><math>~x^0</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ \alpha(10a_0^2) + \sigma^2(- 2a_0^2)   -20n a_0a_2 + 10ma_0^2 </math>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^2</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~\alpha(- 11a_0^2 +20 a_0a_2) + \sigma^2(a_0^2 - 4 a_0a_2) + 60n a_0a_2  -20na_2^2- 25m a_0^2    
+ 20m a_0a_2 + 8n m a_0a_2   
</math><p><math>
- [ 8n(n-1)a_2^2 + 2m(m-1)a_0^2 ]
</math></p>
 </td>
</tr>

<tr>
  <td align="right"><math>~x^4</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ 
\alpha(10a_2^2  - 22 a_0a_2 +3a_0^2) - \sigma^2(2a_2^2- 2a_0a_2 ) - 47n a_0a_2+ 60n a_2^2  - 50m a_0a_2    +11m a_0^2+ 10ma_2^2-12 n m a_0a_2    + 8n m a_2^2
</math><p>
<math>
~ + 16n(n-1)a_2^2 - 4m(m-1)a_0 a_2 + 2m(m-1)a_0^2 
</math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^6</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ 
\alpha(6a_0a_2 - 11a_2^2)  + \sigma^2(a_2^2) + 11 n a_0a_2  +22 m a_0a_2  - 47n a_2^2 - 25m a_2^2  -12 n m a_2^2 + 4n m a_0a_2
</math><p>
<math>~
- 10 n(n-1)a_2^2 -2m(m-1)a_2^2  +4m(m-1)a_0 a_2 
</math></p>
  </td>
</tr>

<tr>
  <td align="right"><math>~x^8</math></td>
  <td align="center">&nbsp; : &nbsp;</td>
  <td align="left">
<math>~ 
a_2[3a_2\alpha  +6a_2 (n + m )  ] + a_2[5ma_2 + 5na_2  + 4n m a_2 ] + [ 2n(n-1)a_2^2 +2m(m-1)a_2^2 ] 
</math>
  </td>
</tr>

</table>
</div>