Editing Appendix/Mathematics/Hypergeometric (section)

=LAWE=

==Familiar Foundation==

Drawing from an [[SSC/Perturbations#2ndOrderODE|accompanying discussion]], we have the,

<div align="center" id="2ndOrderODE">
<font color="#770000">'''LAWE: &nbsp; Linear Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

{{Math/EQ_RadialPulsation01}}
</div>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>g_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- \frac{1}{\rho_0} \frac{dP_0}{dr_0}   \, .</math>
  </td>
</tr>
</table>
Multiplying through by <math>R^2</math>, and making the variable substitutions,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>x</math>
  </td>
  <td align="center">
<math>\rightarrow</math>
  </td>
  <td align="left">
<math>f \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{r_0}{R}</math>
  </td>
  <td align="center">
<math>\rightarrow</math>
  </td>
  <td align="left">
<math>x \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>(4 - 3\gamma_g)</math>
  </td>
  <td align="center">
<math>\rightarrow</math>
  </td>
  <td align="left">
<math>-\alpha \gamma_g \, ,</math>
  </td>
</tr>
</table>
the LAWE may be rewritten as,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>0</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{d^2 f}{dx^2} + \biggl[\frac{4}{x} - \biggl(\frac{g_0 \rho_0 R}{P_0}\biggr) \biggr] \frac{d f}{dx} 
+ \biggl(\frac{\rho_0 R^2}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 - \frac{\alpha \gamma_g g_0}{r_0} \biggr]f
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{d^2 f}{dx^2} + \frac{1}{x}\biggl[4 - \biggl(\frac{g_0 \rho_0 r_0}{P_0}\biggr) \biggr] \frac{d f}{dx} 
+ \biggl[\biggl(\frac{\omega^2\rho_0 R^2}{\gamma_\mathrm{g} P_0} \biggr) 
- \frac{\alpha \gamma_g g_0}{r_0}\biggl(\frac{\rho_0 R^2}{\gamma_\mathrm{g} P_0} \biggr)  \biggr] f
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{d^2 f}{dx^2} + \frac{1}{x}\biggl[4 - \biggl(\frac{g_0 \rho_0 r_0}{P_0}\biggr) \biggr] \frac{d f}{dx} 
+ \biggl[\biggl(\frac{\omega^2\rho_0 R^2}{\gamma_\mathrm{g} P_0} \biggr) 
- \frac{\alpha }{x^2}\biggl(\frac{ g_0 r_0\rho_0 }{ P_0} \biggr)  \biggr] f \, .
</math>
  </td>
</tr>
</table>

If we furthermore adopt the variable definition,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mu</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl(\frac{g_0 \rho_0 r_0}{P_0}\biggr) = - \frac{d\ln P_0}{d\ln r_0} \, ,</math>
  </td>
</tr>
</table>
we obtain what we will refer to as the,

<div align="center" id="Kopal48Expression">
<font color="#770000">'''Kopal (1948) LAWE'''</font><br />
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>0</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{d^2 f}{dx^2} + \frac{(4-\mu)}{x} \cdot \frac{df}{dx} 
+ \biggl[\biggl(\frac{\omega^2\rho_0 R^2}{\gamma_\mathrm{g} P_0} \biggr) 
- \frac{\alpha \mu}{x^2} \biggr]f \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Kopal48 }}, p. 378, Eq. (6)<br />
{{ VdBorght70 }}, p. 325, Eq. (1)
  </td>
</tr>

</table>
</div>

==Specifically for Polytropes==

Let's look at the expression for the function, <math>\mu</math>, that arises in the context of polytropic spheres. 

===General Expression for the Function &mu;===
First, we note that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>r_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>a \xi \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\rho_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_c \theta^n \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K\biggl[\rho_c \theta^n\biggr]^{(n+1)/n} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>M_r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\pi a^3\rho_c \biggl( - \xi^2 \frac{d\theta}{d\xi}\biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>g_0 \equiv \frac{GM_r}{r_0^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\pi G a\rho_c \biggl( - \frac{d\theta}{d\xi}\biggr) \, ,</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>a</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl[\frac{(n+1)K}{4\pi G}\biggr]^{1 /  2} \rho_c^{(1-n)/2n} \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ K</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{4\pi G}{(n+1)} \biggr] a^2 \rho_c^{(n-1)/n} \, .</math>
  </td>
</tr>
</table>
Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mu = \biggl(\frac{g_0 \rho_0 r_0}{P_0}\biggr)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
4\pi G a\rho_c \biggl( - \frac{d\theta}{d\xi}\biggr)
\rho_c \theta^n a\xi
\biggl[\rho_c \theta^n\biggr]^{-(n+1)/n}\biggl[ \frac{(n+1)}{4\pi G} \biggr] a^{-2} \rho_c^{-(n-1)/n}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
(n+1)\biggl( - \frac{d\theta}{d\xi}\biggr)
\theta^n \xi
\theta^{-(n+1)} \rho_c^{2-(n+1)/n-(n-1)/n}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
(n+1)\biggl( - \frac{\xi}{\theta}\cdot \frac{d\theta}{d\xi}\biggr) 
=
(n+1)\biggl( - \frac{d\ln \theta}{d\ln \xi}\biggr) 
\, .
</math>
  </td>
</tr>
</table>
Alternatively,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mu= - \frac{d\ln P_0}{d\ln r_0}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\frac{r_0}{P_0} \cdot \frac{dP_0}{d r_0}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\xi \theta^{-(n+1)}\cdot \frac{d}{d\xi} \biggl[\theta^{(n+1)}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-(n+1) \biggl(\frac{\xi}{\theta}\cdot \frac{d\theta}{d\xi} \biggr)
=
(n+1)\biggl( - \frac{d\ln \theta}{d\ln \xi}\biggr) 
\, .
</math>
  </td>
</tr>
</table>
Yes!

===Trial Displacement Function===
Now, building on an [[SSC/Stability/InstabilityOnsetOverview#Analyses_of_Radial_Oscillations|accompanying discussion]], let's guess,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>f_\mathrm{trial}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3(n-1)}{2n} \biggl[ 1 + \biggl( \frac{n-3}{n-1} \biggr)\biggl(\frac{1}{\xi \theta^n}\biggr)\frac{d\theta}{d\xi} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl[ \frac{2n}{3(n-1)} \biggr] f_\mathrm{trial}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 + \biggl( \frac{n-3}{n-1} \biggr)\xi^{-1} \theta^{-n}\biggl[ \frac{\theta}{\xi} \cdot \frac{d\ln\theta}{d\ln\xi} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 + \biggl( \frac{3-n}{n-1} \biggr)\xi^{-2} \theta^{(1-n)}\cdot \biggl( - \frac{d\ln\theta}{d\ln\xi} \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 + \biggl[ \frac{3-n}{(n+1)(n-1)} \biggr]\xi^{-2} \theta^{(1-n)}\cdot \mu
</math>
  </td>
</tr>
</table>

Flipping it around, we have alternatively,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mu</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{(n+1)(n-1)}{3-n} \biggr]\biggl\{ \biggl[ \frac{2n}{3(n-1)} \biggr] f_\mathrm{trial} - 1 \biggr\}
\xi^{2} \theta^{(n-1)}
</math>
  </td>
</tr>
</table>



===Plug into Kopal (1948) LAWE===

====Replace f<sub>trial</sub> by &mu;====
Plugging this trial function into the Kopal (1948) LAWE and recognizing that <math>x = \xi/\xi_1</math>, we find,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>\xi_1^{-2} \cdot ~\mathrm{LAWE}</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{d^2 f_\mathrm{trial}}{d\xi^2} + \frac{(4-\mu)}{\xi} \cdot \frac{df_\mathrm{trial}}{d\xi} 
+ \biggl[\biggl(\frac{\omega^2\rho_0 R^2}{\gamma_\mathrm{g} P_0 \xi_1^2} \biggr) 
- \frac{\alpha \mu}{\xi^2} \biggr]f_\mathrm{trial} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl[ \frac{2n}{3(n-1)} \biggr] \xi_1^{-2} \cdot ~\mathrm{LAWE}</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{d^2 }{d\xi^2} \biggl\{ 1 + \biggl[ \frac{3-n}{(n+1)(n-1)} \biggr]\xi^{-2} \theta^{(1-n)}\cdot \mu \biggr\}
+ \frac{(4-\mu)}{\xi} \cdot \frac{d}{d\xi} \biggl\{ 1 + \biggl[ \frac{3-n}{(n+1)(n-1)} \biggr]\xi^{-2} \theta^{(1-n)}\cdot \mu \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td> 
  <td align="center">
&nbsp;
  </td> 
  <td align="left">
<math>
+ \biggl[\biggl(\frac{\omega^2\rho_0 R^2}{\gamma_\mathrm{g} P_0 \xi_1^2} \biggr) 
- \frac{\alpha \mu}{\xi^2} \biggr]\biggl\{ 1 + \biggl[ \frac{3-n}{(n+1)(n-1)} \biggr]\xi^{-2} \theta^{(1-n)}\cdot \mu \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl[ \frac{2n(n+1)}{3} \biggr] \xi_1^{-2} \cdot ~\mathrm{LAWE}</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{d^2 }{d\xi^2} \biggl\{ (3-n) \xi^{-2} \theta^{(1-n)}\cdot \mu \biggr\}
+ \frac{(4-\mu)}{\xi} \cdot \frac{d}{d\xi} \biggl\{ (3-n) \xi^{-2} \theta^{(1-n)}\cdot \mu \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td> 
  <td align="center">
&nbsp;
  </td> 
  <td align="left">
<math>
+ \biggl[\biggl(\frac{\omega^2\rho_0 R^2}{\gamma_\mathrm{g} P_0 \xi_1^2} \biggr) 
- \frac{\alpha \mu}{\xi^2} \biggr]\biggl\{ (n+1)(n-1) + (3-n)\xi^{-2} \theta^{(1-n)}\cdot \mu \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl[ \frac{2n(n+1)}{3(3-n)} \biggr] \xi_1^{-2} \cdot ~\mathrm{LAWE}</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{d^2 }{d\xi^2} \biggl\{ \xi^{-2} \theta^{(1-n)}\cdot \mu \biggr\}
+ \frac{(4-\mu)}{\xi} \cdot \frac{d}{d\xi} \biggl\{ \xi^{-2} \theta^{(1-n)}\cdot \mu \biggr\}
+ \biggl[\biggl(\frac{\omega^2\rho_0 R^2}{\gamma_\mathrm{g} P_0 \xi_1^2} \biggr) 
- \frac{\alpha \mu}{\xi^2} \biggr]\biggl\{ \frac{(n+1)(n-1)}{(3-n)} + \xi^{-2} \theta^{(1-n)}\cdot \mu \biggr\} 
\, .
</math>
  </td>
</tr>
</table>
Noting that, <math>R/\xi_1 = a</math> and
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>\frac{\rho_0}{P_0}</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\rho_c \theta^n K^{-1} \rho_c^{-(n+1)/n}\theta^{-(n+1)}
=
K^{-1}\rho_c^{-1/n} \theta^{-1} \, ,
</math>
  </td>
</tr>
</table>
the frequency-squared term may be rewritten as,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>\frac{\omega^2}{\gamma_g}\biggl(\frac{\rho_0 a^2}{ P_0 } \biggr)</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\biggl( \frac{\omega^2}{\gamma_g} \biggr)
K^{-1}\rho_c^{-1/n} \theta^{-1}
\biggl[\frac{(n+1)K}{4\pi G}\biggr] \rho_c^{(1-n)/n}
=
\biggl( \frac{\omega^2}{\gamma_g} \biggr)
\biggl[\frac{(n+1)}{4\pi \rho_c G}\biggr]\theta^{-1} \, .
</math>
  </td>
</tr>
</table>

====Replace &mu; by f<sub>trial</sub>====
Making instead the alternate substitution, namely,

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

<tr>
  <td align="right">
<math>\mu</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{(n+1)(n-1)}{3-n} \biggr]\biggl\{ \biggl[ \frac{2n}{3(n-1)} \biggr] f_\mathrm{trial} - 1 \biggr\}
\xi^{2} \theta^{(n-1)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl\{ 
\frac{2n(n+1)}{3(3-n)} \cdot f_\mathrm{trial} 
- \frac{(n+1)(n-1)}{3-n} 
\biggr\}
\xi^{2} \theta^{(n-1)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{3(3-n)}
\biggl[  \underbrace{2n(n+1)}_{A} f_\mathrm{trial} - \underbrace{3(n+1)(n-1)}_{B} \biggr]\xi^{2} \theta^{(n-1)} 
</math>
  </td>
</tr>
</table>

we have,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>\xi_1^{-2} \cdot ~\mathrm{LAWE}</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{d^2 f_\mathrm{trial}}{d\xi^2} 
+ \biggl\{\frac{4}{\xi} \biggr\} \frac{df_\mathrm{trial}}{d\xi} 
- \biggl\{ \frac{\mu}{\xi} \biggr\} \frac{df_\mathrm{trial}}{d\xi} 
+ 
\biggl( \frac{\omega^2}{\gamma_g} \biggr)\biggl[\frac{(n+1)}{4\pi \rho_c G}\biggr]\frac{f_\mathrm{trial}}{\theta} 
- \biggl(\frac{\alpha}{\xi^2}  \biggr) \mu f_\mathrm{trial}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl[ \frac{3(3-n)  }{\xi_1^2} \biggr] ~\mathrm{LAWE}</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
3(3-n)\frac{d^2 f_\mathrm{trial}}{d\xi^2} 
+ \biggl\{\frac{12(3-n)}{\xi} \biggr\} \frac{df_\mathrm{trial}}{d\xi} 
- \biggl[  A f_\mathrm{trial} - B \biggr]\xi \theta^{(n-1)}\frac{df_\mathrm{trial}}{d\xi} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td> 
  <td align="center">
&nbsp;
  </td> 
  <td align="left">
<math>
+ 
3(3-n)\biggl( \frac{\omega^2}{\gamma_g} \biggr)\biggl[\frac{(n+1)}{4\pi \rho_c G}\biggr]\frac{f_\mathrm{trial}}{\theta} 
- \alpha \biggl[  A f_\mathrm{trial} - B \biggr]\theta^{(n-1)} f_\mathrm{trial} \, .
</math>
  </td>
</tr>
</table>
Noting that,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>\frac{d}{d\xi} \biggl[ \xi \theta^{(n-1)}f_\mathrm{trial} \biggr]</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\biggl\{
\theta^{(n-1)}f_\mathrm{trial} 
+ \xi f_\mathrm{trial} (n-1)\theta^{(n-2)}\frac{d\theta}{d\xi}
+ \xi\theta^{(n-1)}\frac{df_\mathrm{trial}}{d\xi}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ 
\xi\theta^{(n-1)}\frac{df_\mathrm{trial}}{d\xi}
</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{d}{d\xi} \biggl[ \xi \theta^{(n-1)}f_\mathrm{trial} \biggr]
-
\biggl[
\theta^{(n-1)}f_\mathrm{trial} 
+ \xi f_\mathrm{trial} (n-1)\theta^{(n-2)}\frac{d\theta}{d\xi}
\biggr] 
\, ,
</math>
  </td>
</tr>
</table>
we furthermore can write,

<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \biggl[ \frac{3(3-n)  }{\xi_1^2} \biggr] ~\mathrm{LAWE}</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
3(3-n)\frac{d^2 f_\mathrm{trial}}{d\xi^2} 
+ \biggl\{\frac{12(3-n)}{\xi} \biggr\} \frac{df_\mathrm{trial}}{d\xi} 
+ 3(3-n)\biggl( \frac{\omega^2}{\gamma_g} \biggr)\biggl[\frac{(n+1)}{4\pi \rho_c G}\biggr]\frac{f_\mathrm{trial}}{\theta} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td> 
  <td align="center">
&nbsp;
  </td> 
  <td align="left">
<math>
- \alpha \biggl[  A f_\mathrm{trial} - B \biggr]\theta^{(n-1)} f_\mathrm{trial} 
- \biggl[  A f_\mathrm{trial} - B \biggr]
\biggl\{
\frac{d}{d\xi} \biggl[ \xi \theta^{(n-1)}f_\mathrm{trial} \biggr]
-
\biggl[
\theta^{(n-1)}f_\mathrm{trial} 
+ \xi f_\mathrm{trial} (n-1)\theta^{(n-2)}\frac{d\theta}{d\xi}
\biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
3(3-n)\frac{d^2 f_\mathrm{trial}}{d\xi^2} 
+ \biggl\{\frac{12(3-n)}{\xi} \biggr\} \frac{df_\mathrm{trial}}{d\xi} 
+ 3(3-n)\biggl( \frac{\omega^2}{\gamma_g} \biggr)\biggl[\frac{(n+1)}{4\pi \rho_c G}\biggr]\frac{f_\mathrm{trial}}{\theta} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td> 
  <td align="center">
&nbsp;
  </td> 
  <td align="left">
<math>
- \biggl[  A f_\mathrm{trial} - B \biggr]
\biggl\{
\frac{d}{d\xi} \biggl[ \xi \theta^{(n-1)}f_\mathrm{trial} \biggr]
-
\biggl[
\theta^{(n-1)}f_\mathrm{trial} 
+ \xi f_\mathrm{trial} (n-1)\theta^{(n-2)}\frac{d\theta}{d\xi}
\biggr] + \alpha \theta^{(n-1)} f_\mathrm{trial}
\biggr\}
</math>
  </td>
</tr>
</table>