Editing SSC/Stability/NeutralMode (section)

==Neutral Mode==
Again, adopting the ''trial'' eigenfunction,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right"><math>x_t</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
a - b\Delta
\, ,
</math>
</td>
</tr>
</table>
from the,

<div align="center" id="Delta_Highlighted">
<font color="#770000">'''&Delta;-Highlighted LAWE'''</font><br />

{{Math/EQ_RadialPulsation04}}
</div>
we can write,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right"><math>\mathrm{LAWE}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- b \cdot \frac{d^2\Delta}{dr_0^2} - \frac{b}{r_0}\biggl[4 + \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{d\Delta}{dr_0} 
+ \alpha\biggl[ \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{(a - b\Delta)}{r_0^2} 
- \frac{1}{\Delta} \biggl[\frac{d\ln P_0}{d\ln r_0}\cdot \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \frac{(a - b\Delta)}{r_0^2} 
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- r_0^2 \cdot \frac{d^2\Delta}{dr_0^2} - r_0\biggl[4 + \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{d\Delta}{dr_0} 
+ \alpha\biggl[ \frac{d\ln P_0}{d\ln r_0} \biggr] \biggl( \frac{a}{b} - \Delta \biggr) 
- \frac{1}{\Delta} \biggl[\frac{d\ln P_0}{d\ln r_0}\cdot \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr)\biggr] \biggl(\frac{a}{b} - \Delta \biggr) 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- r_0^2 \cdot \frac{d^2\Delta}{dr_0^2} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
- r_0\biggl[4 + \frac{d\ln P_0}{d\ln r_0} \biggr] \frac{d\Delta}{dr_0} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \alpha\biggl[ \frac{d\ln P_0}{d\ln r_0} \biggr] \biggl( \frac{a}{b} - \Delta \biggr) 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
\biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]  
- \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- \frac{4}{15} \biggl(\frac{r_0}{R}\biggr)^2 
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} 
\biggl[1 + 3 \biggl(\frac{r_0}{R}\biggr)^2 \biggr]
 </math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
- \frac{16}{15}\biggl(\frac{r_0}{R}\biggr)^2  
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2} 
- \frac{4}{15}\biggl(\frac{r_0}{R}\biggr)^2 
\biggl\{ -5 \biggl(\frac{r_0}{R}\biggr)^2
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}
 \biggr\} 
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-2} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \alpha
\biggl\{ -5 \biggl(\frac{r_0}{R}\biggr)^2
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}
\biggr\} 
\biggl\{ \frac{a}{b} - \Delta \biggr\} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
\biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]  
- \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- \frac{4}{3}\biggl(\frac{r_0}{R}\biggr)^2\biggl[ 1 - \frac{1}{5} \biggl(\frac{r_0}{R}\biggr)^2 \biggr]
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3}
 </math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+~ \frac{4}{3}\biggl(\frac{r_0}{R}\biggr)^4 
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ ~ \frac{5\alpha}{3}
\biggl(\frac{r_0}{R}\biggr)^2 \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-2} 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} 
\biggl\{
1 - \frac{3a}{b} -\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2
\biggr\} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
\biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]  
- \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- \frac{4}{3}\biggl(\frac{r_0}{R}\biggr)^2\biggl[ 1 - \frac{1}{5} \biggl(\frac{r_0}{R}\biggr)^2 \biggr]
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3}
+ ~ \frac{5\alpha}{3}
\biggl(\frac{r_0}{R}\biggr)^2 \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-2} 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} 
\biggl(1 - \frac{3a}{b} \biggr) 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+~ \frac{4}{3}\biggl(\frac{r_0}{R}\biggr)^4 
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3} 
- ~ \alpha
\biggl(\frac{r_0}{R}\biggr)^4 \biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-2} 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
\biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]  
- \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} 
</math>
</td>
</tr>
</table>

Continuing &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right"><math>\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
\biggl(\frac{r_0}{R}\biggr)^2 \biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3}

\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}
\biggl\{
\frac{5\alpha}{3}\biggl(1 - \frac{3a}{b} \biggr)
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2 \biggr]
\biggl[1-\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
 
- \frac{4}{3}\biggl[ 1 - \frac{1}{5} \biggl(\frac{r_0}{R}\biggr)^2 \biggr]\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+~ \biggl(\frac{4}{3} - \alpha  \biggr)\biggl(\frac{r_0}{R}\biggr)^4\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3}
 </math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
-~ \alpha \biggl(\frac{r_0}{R}\biggr)^6\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3}
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
\biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]  
- \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
\biggl(\frac{r_0}{R}\biggr)^2 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3}
\biggl\{
\frac{5\alpha}{3}\biggl(1 - \frac{3a}{b} \biggr)
\biggl[1-\frac{8}{5}\biggl(\frac{r_0}{R}\biggr)^2 + \frac{3}{5} \biggl( \frac{r_0}{R}\biggr)^4\biggr]
- 
~\frac{4}{3}\biggl[ 1 - \frac{7}{5} \biggl(\frac{r_0}{R}\biggr)^2 + \frac{1}{10}\biggl(\frac{r_0}{R}\biggr)^4\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+~ \biggl(\frac{4}{3} - \alpha  \biggr)\biggl(\frac{r_0}{R}\biggr)^2
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
-~ \alpha \biggl(\frac{r_0}{R}\biggr)^4\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
\biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]  
- \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
\biggl(\frac{r_0}{R}\biggr)^2 
\biggl[1-\frac{1}{2}\biggl(\frac{r_0}{R}\biggr)^2\biggr]^{-1}
\biggl[ 1 - \biggl(\frac{r_0}{R} \biggr)^2 \biggr]^{-3}
\biggl\{ \biggl[\frac{5\alpha}{3}\biggl(1 - \frac{3a}{b} \biggr) - \frac{4}{3} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \biggl(\frac{r_0}{R}\biggr)^2
\biggl[
-\frac{8\alpha}{3}\biggl(1 - \frac{3a}{b} \biggr) + \frac{28}{15}
\biggr]
+ \biggl( \frac{r_0}{R}\biggr)^4 \biggl[\alpha\biggl(1 - \frac{3a}{b} \biggr)
- ~\frac{2}{15} \biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+~ \biggl(\frac{4}{3} - \alpha  \biggr)\biggl(\frac{r_0}{R}\biggr)^2
\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
-~ \alpha \biggl(\frac{r_0}{R}\biggr)^4\biggl[1-\frac{3}{5}\biggl(\frac{r_0}{R}\biggr)^2\biggr] 
\biggr\}
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \frac{\rho_c}{\rho_0} \biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
\biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]  
- \frac{a}{b}\biggl[\frac{1}{\Delta} \cdot \frac{d\ln P_0}{d\ln r_0}\biggr] \biggr\} 
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right"><math>\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- r_0^2 \cdot \frac{d^2\Delta}{dr_0^2} 
- r_0\biggl[4 \biggr] \frac{d\Delta}{dr_0} 
+ \biggl[\frac{d\ln P_0}{d\ln r_0} \biggr]\biggl[ - r_0\cdot \frac{d\Delta}{dr_0} + \alpha \biggl( \frac{a}{b} - \Delta \biggr)\biggr]
+ \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) 
\biggl\{1  - \frac{a}{b}\biggl[\frac{1}{\Delta} \biggr] \biggr\} \frac{\rho_c}{\rho_0} 
\, ,
</math>
</td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right"><math>\frac{\rho_0}{\rho_c}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
\biggl(1 - x^2\biggr) \, ,
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Delta</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
\frac{1}{3} \biggl(1 - \frac{3}{5} x^2\biggr)\biggl(1 - x^2\biggr)^{-1} \, ,
</math>
</td>
</tr>

<tr>
<td align="right"><math>\frac{d\ln P_0}{d\ln r_0}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
-5x^2  \biggl(1 - \frac{3}{5} x^2\biggr)\biggl(1 - x^2\biggr)^{-1} \biggl(1 - \frac{1}{2}x^2\biggr)^{-1}
=
-15 x^2 \biggl(1 - \frac{1}{2}x^2\biggr)^{-1} \Delta \, ,
</math>
</td>
</tr>

<tr>
<td align="right"><math>r_0 \cdot \frac{d\Delta}{dr_0}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
\frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-2} \, ,
</math>
</td>
</tr>

<tr>
<td align="right"><math>r_0^2 \cdot \frac{d^2\Delta}{dr_0^2}</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
\frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-3}\biggl( 1 + 3x^2\biggr) \, .
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right"><math>\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- r_0^2 \cdot \frac{d^2\Delta}{dr_0^2} 
- r_0\biggl[4 \biggr] \frac{d\Delta}{dr_0} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \biggl[\frac{d\ln P_0}{d\ln r_0} \biggr]\biggl[ - r_0\cdot \frac{d\Delta}{dr_0} + \alpha \biggl( \frac{a}{b} - \Delta \biggr)\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
1  
- \frac{a}{b}\biggl[\frac{1}{\Delta} \biggr] \biggr\} \frac{\rho_c}{\rho_0} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- \frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-3}\biggl( 1 + 3x^2\biggr)  
- \frac{16}{15} x^2 \biggl(1 - x^2\biggr)^{-2} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \biggl[\frac{d\ln P_0}{d\ln r_0} \biggr]
\biggl[\frac{a\alpha}{b} - \frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-2} 
- \frac{\alpha}{3} \biggl(1 - \frac{3}{5} x^2\biggr)\biggl(1 - x^2\biggr)^{-1}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+ \biggl[\frac{d\ln P_0}{d\ln r_0}\biggr]\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
1  
- \frac{3a}{b}\biggl(1 - \frac{3}{5}x^2\biggr)^{-1} \biggl(1-x^2\biggr)\biggr\} \biggl(1 - x^2\biggr)^{-1} \, .
</math>
</td>
</tr>
</table>

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

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{1}{x^2}\biggl(1 - x^2\biggr)
\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- \frac{4}{15} \biggl(1 - x^2\biggr)^{-2}\biggl( 1 + 3x^2\biggr)  
- \frac{16}{15} \biggl(1 - x^2\biggr)^{-1} 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
-5 \biggl(1 - \frac{3}{5} x^2\biggr) \biggl(1 - \frac{1}{2}x^2\biggr)^{-1}
\biggl[\frac{a\alpha}{b} - \frac{4}{15} x^2 \biggl(1 - x^2\biggr)^{-2} 
- \frac{\alpha}{3} \biggl(1 - \frac{3}{5} x^2\biggr)\biggl(1 - x^2\biggr)^{-1}\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
-5 \biggl(1 - \frac{3}{5} x^2\biggr) \biggl(1 - \frac{1}{2}x^2\biggr)^{-1}\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
1  
- \frac{3a}{b}\biggl(1 - \frac{3}{5}x^2\biggr)^{-1} \biggl(1-x^2\biggr)\biggr\} \biggl(1 - x^2\biggr)^{-1}
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{1}{x^2}\biggl(1 - x^2\biggr)^3
\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math> 
- \frac{4}{15} \biggl( 1 + 3x^2\biggr)  
- \frac{16}{15} \biggl(1 - x^2\biggr) 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
-5 \biggl(1 - \frac{3}{5} x^2\biggr) \biggl(1 - \frac{1}{2}x^2\biggr)^{-1}
\biggl[\frac{a\alpha}{b}\biggl(1 - x^2\biggr)^2 - \frac{4}{15} x^2  
- \frac{\alpha}{3} \biggl(1 - \frac{3}{5} x^2\biggr)\biggl(1 - x^2\biggr)\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
-5 \biggl(1 - \frac{1}{2}x^2\biggr)^{-1}\biggl( \frac{\sigma_c^2}{6\gamma_g}\biggr) \biggl\{
\biggl(1 - \frac{3}{5} x^2\biggr)   
- \frac{3a}{b} \biggl(1-x^2\biggr)\biggr\} \biggl(1 - x^2\biggr)
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ \frac{15}{x^2}\biggl(1 - x^2\biggr)^3 \biggl(1 - \frac{1}{2}x^2\biggr)
\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>- 2 \biggl(2 - x^2\biggr)\biggl(5 - x^2\biggr)
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
-~ \biggl(5 - 3 x^2\biggr) 
\biggl[\frac{15a\alpha}{b}\biggl(1 - 2x^2 + x^4\biggr) - 4 x^2  
- \alpha \biggl(5 - 8 x^2 + 3x^4 \biggr)\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
-~ \biggl( \frac{5\sigma_c^2}{2\gamma_g}\biggr) \biggl\{
\biggl(5 - 3 x^2\biggr)   
- \frac{15a}{b} \biggl(1-x^2\biggr)\biggr\} \biggl(1 - x^2\biggr)
</math>
</td>
</tr>

<tr>
<td align="right"><math>\Rightarrow ~~~ -~\frac{15}{x^2}\biggl(1 - x^2\biggr)^3 \biggl(1 - \frac{1}{2}x^2\biggr)
\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
20 - 14x^2 + 2x^4
+~ \biggl(5 - 3 x^2\biggr) 
\biggl[ 
\biggl(\frac{15a\alpha}{b}-5\alpha \biggr) + x^2\biggl( \frac{30 a\alpha}{b} -4 + 8\alpha \biggr) + \alpha x^4 \biggl(\frac{15a}{b} - 3 \biggr)
\biggr]
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="left">
<math> 
+~ \biggl( \frac{5\sigma_c^2}{2\gamma_g}\biggr) \biggl[
\biggl(5 - \frac{15a}{b}\biggr) + x^2 \biggl(-3 + \frac{15a}{b}  \biggr) \biggr] \biggl(1 - x^2\biggr) 
\, .
</math>
</td>
</tr>
</table>
Now, if we set <math>(15a/b) = 3</math>, this last expression reduces to,

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

<tr>
<td align="right"><math>-~\frac{15}{x^2}\biggl(1 - x^2\biggr)^3 \biggl(1 - \frac{1}{2}x^2\biggr)
\frac{1}{b} \biggl[ r_0^2 \times \mathrm{LAWE} \biggr]</math></td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
20 - 14x^2 + 2x^4
-~ 2\biggl(5 - 3 x^2\biggr) 
\biggl[ 
\alpha + x^2 (2 - 7\alpha ) 
\biggr]
+~ \biggl( \frac{5\sigma_c^2}{\gamma_g}\biggr) \biggl(1 - x^2\biggr) 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>
20 - 14x^2 + 2x^4
-~ 2
\biggl[ 5\alpha + x^2 (10-38\alpha ) + x^4(21\alpha - 6) \biggr]
+~ \biggl( \frac{5\sigma_c^2}{\gamma_g}\biggr) \biggl(1 - x^2\biggr) 
</math>
</td>
</tr>

<tr>
<td align="right">&nbsp;</td>
<td align="center"><math>=</math></td>
<td align="left">
<math>x^0 \biggl[20 - 10\alpha +  \biggl( \frac{5\sigma_c^2}{\gamma_g}\biggr) \biggr]
+ x^2\biggl[76\alpha -34  - \biggl( \frac{5\sigma_c^2}{\gamma_g}\biggr) \biggr]
+ 14 x^4 \biggl[1 - 3\alpha \biggr]
</math>
</td>
</tr>
</table>