Editing SSC/Structure/OtherAnalyticRamblings (section)

====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.