Editing SSC/Structure/OtherAnalyticRamblings (section)

====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>