Editing SSC/Stability/n1PolytropeLAWE/Pt2 (section)

====Development====
Multiplying through the [[#Simplistic_Layout|above LAWE]] by <math>~(x \xi^{-3})</math> 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>~
\frac{\sin\xi }{\xi} \cdot  \frac{d^2x}{d\xi^2} + 2 \biggl[\frac{ \sin\xi +  \xi \cos \xi }{\xi^2}\biggr] \frac{dx}{d\xi} + 
\biggl[ \sigma^2   - 2\alpha \biggl( \frac{\sin\xi - \xi \cos \xi}{\xi^3} \biggr) \biggr]x
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\frac{d}{d\xi}\biggl[\frac{\sin\xi}{\xi}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{\sin\xi}{\xi^2} + \frac{\cos\xi}{\xi}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{\xi\cos\xi - \sin\xi }{\xi^2} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
And, hence,
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<tr>
  <td align="right">
<math>~\frac{d^2}{d\xi^2}\biggl[\frac{\sin\xi}{\xi}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d}{d\xi}\biggl[ \frac{\cos\xi }{\xi} -  \frac{\sin\xi }{\xi^2}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
 -\frac{\cos\xi}{\xi^2} -\frac{\sin\xi}{\xi} + \frac{2\sin\xi}{\xi^3} - \frac{\cos\xi}{\xi^2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\frac{\sin\xi}{\xi} + 2\biggl[ \frac{\sin\xi -\xi\cos\xi}{\xi^3} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
So, we can write,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2}{d\xi^2} \biggl\{ \biggl( \frac{\sin\xi}{\xi}\biggr)x \biggr\}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{d}{d\xi} \biggl\{ 
 \biggl(\frac{\sin\xi}{\xi}\biggr)\frac{dx}{d\xi}  + x\frac{d}{d\xi} \biggl[ \biggl(\frac{\sin\xi}{\xi}\biggr) \biggr] 
\biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\sin\xi}{\xi} \cdot \frac{d^2 x}{d\xi^2}
+ 2\frac{dx}{d\xi} \cdot \biggl[\frac{d}{d\xi}\biggr(\frac{\sin\xi}{\xi}\biggr)  \biggr]
+ x \cdot \frac{d^2}{d\xi^2} \biggl(\frac{\sin\xi}{\xi}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\sin\xi}{\xi} \cdot \frac{d^2 x}{d\xi^2}
+ 2\frac{dx}{d\xi} \cdot \biggl[ \frac{\xi\cos\xi - \sin\xi }{\xi^2} \biggr]
+ x \cdot \biggl\{ -\frac{\sin\xi}{\xi} + 2\biggl[ \frac{\sin\xi -\xi\cos\xi}{\xi^3} \biggr] \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>
This means that we can rewrite the LAWE as,

<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>~
\frac{d^2}{d\xi^2} \biggl\{ \biggl( \frac{\sin\xi}{\xi}\biggr)x \biggr\} - 2\frac{dx}{d\xi} \cdot \biggl[ \frac{\xi\cos\xi - \sin\xi }{\xi^2} \biggr]
- x \cdot \biggl\{ -\frac{\sin\xi}{\xi} + 2\biggl[ \frac{\sin\xi -\xi\cos\xi}{\xi^3} \biggr] \biggr\}
+ 2 \biggl[\frac{ \sin\xi +  \xi \cos \xi }{\xi^2}\biggr] \frac{dx}{d\xi} + 
\biggl[ \sigma^2   - 2\alpha \biggl( \frac{\sin\xi - \xi \cos \xi}{\xi^3} \biggr) \biggr]x
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2}{d\xi^2} \biggl\{ \biggl( \frac{\sin\xi}{\xi}\biggr)x \biggr\} 
+ 4 \biggl[\frac{ \sin\xi  }{\xi^2}\biggr] \frac{dx}{d\xi} 
+ \biggl\{ \frac{\sin\xi}{\xi} 
+ \sigma^2   - 2(1+\alpha) \biggl( \frac{\sin\xi - \xi \cos \xi}{\xi^3} \biggr) \biggr\} \cdot x \, .
</math>
  </td>
</tr>
</table>
</div>

We recognize, also, that,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{\xi} \cdot \frac{d}{d\xi}\biggl[ \biggl(\frac{\sin\xi}{\xi} \biggr) x \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \frac{\xi\cos\xi - \sin\xi }{\xi^3} \biggr]x + \biggl(\frac{\sin\xi}{\xi^2} \biggr)\frac{dx}{d\xi} \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ 
4\biggl(\frac{\sin\xi}{\xi^2} \biggr)\frac{dx}{d\xi} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4}{\xi} \cdot \frac{d}{d\xi}\biggl[ \biggl(\frac{\sin\xi}{\xi} \biggr) x \biggr]
+ 4\biggl[ \frac{\sin\xi - \xi\cos\xi }{\xi^3} \biggr]x
\, .
</math>
  </td>
</tr>
</table>
</div>
So the LAWE becomes,

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<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>~
\frac{d^2}{d\xi^2} \biggl\{ \biggl( \frac{\sin\xi}{\xi}\biggr)x \biggr\} 
+ \frac{4}{\xi} \cdot \frac{d}{d\xi}\biggl[ \biggl(\frac{\sin\xi}{\xi} \biggr) x \biggr]
+ 4\biggl[ \frac{\sin\xi - \xi\cos\xi  }{\xi^3} \biggr]x
+ \biggl\{ \frac{\sin\xi}{\xi} 
+ \sigma^2   - 2(1+\alpha) \biggl( \frac{\sin\xi - \xi \cos \xi}{\xi^3} \biggr) \biggr\} \cdot x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2}{d\xi^2} \biggl\{ \biggl( \frac{\sin\xi}{\xi}\biggr)x \biggr\} 
+ \frac{4}{\xi} \cdot \frac{d}{d\xi}\biggl[ \biggl(\frac{\sin\xi}{\xi} \biggr) x \biggr]
+ \biggl\{ \frac{\sin\xi}{\xi} 
+ \sigma^2   + [4- 2(1+\alpha)] \biggl( \frac{\sin\xi - \xi \cos \xi}{\xi^3} \biggr) \biggr\} \cdot x 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2 \Upsilon}{d\xi^2} 
+ \frac{4}{\xi} \cdot \frac{d\Upsilon}{d\xi}
+ \Upsilon 
+ \biggl[ \sigma^2   + 2(1-\alpha) \biggl( \frac{\sin\xi - \xi \cos \xi}{\xi^3} \biggr) \biggr] \cdot x \, ,
</math>
  </td>
</tr>
</table>
</div>
where we have introduced the new, modified eigenfunction,
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<math>\Upsilon \equiv \biggl( \frac{\sin\xi}{\xi} \biggr) x \, .</math>
</div>

Alternatively, the LAWE may be written as,

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<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>~
\frac{d^2 \Upsilon}{d\xi^2} 
+ \frac{4}{\xi} \cdot \frac{d\Upsilon}{d\xi}
+ \biggl[ \sigma^2   + 2(1-\alpha) \biggl( \frac{\sin\xi - \xi \cos \xi}{\xi^3} \biggr) + \frac{\sin\xi}{\xi} \biggr] \cdot x \, ;
</math>
  </td>
</tr>
</table>
</div>
or,

<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>~
\frac{\xi^2}{\Upsilon} \cdot \frac{d^2 \Upsilon}{d\xi^2} 
+ \frac{4\xi}{\Upsilon} \cdot \frac{d\Upsilon}{d\xi}
+ \biggl[ \sigma^2   + 2(1-\alpha) \biggl( \frac{\sin\xi - \xi \cos \xi}{\xi^3} \biggr) + \frac{\sin\xi}{\xi} \biggr] \cdot \frac{\xi^3}{\sin\xi} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\xi^2}{\Upsilon} \cdot \frac{d^2 \Upsilon}{d\xi^2} 
+ \frac{4\xi}{\Upsilon} \cdot \frac{d\Upsilon}{d\xi}
+ \biggl[ \sigma^2 \biggl(\frac{\xi^3}{\sin\xi} \biggr)   + 2(1-\alpha) \biggl( 1 - \xi \cot \xi \biggr) + \xi^2 \biggr]   
</math>
  </td>
</tr>
</table>
</div>


Now, if we adopt the homentropic convention that arises from setting, <math>~\gamma = (n+1)/n</math>, then for our <math>~n=1</math> polytropic configuration, we should set, <math>~\gamma = 2</math> and, hence, <math>~\alpha = 1</math>.  This will mean that the lat term in this LAWE naturally goes to zero.  Hence, we have,

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

<tr>
  <td align="right">
<math>~- \sigma^2 x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{d^2 \Upsilon}{d\xi^2} 
+ \frac{4}{\xi} \cdot \frac{d\Upsilon}{d\xi} + \Upsilon
\, ;
</math>
  </td>
</tr>
</table>
</div>
or,

<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>~
\frac{d^2 \Upsilon}{d\xi^2} 
+ \frac{4}{\xi} \cdot \frac{d\Upsilon}{d\xi} + \biggl[1 + \sigma^2 \biggl(\frac{\xi}{\sin\xi}\biggr) \biggr] \Upsilon \, ;
</math>
  </td>
</tr>
</table>
</div>
or,

<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>~
\frac{\xi^2}{\Upsilon} \cdot \frac{d^2 \Upsilon}{d\xi^2} 
+ \frac{4\xi}{\Upsilon} \cdot \frac{d\Upsilon}{d\xi} + \biggl[\xi^2 + \sigma^2 \biggl(\frac{\xi^3}{\sin\xi}\biggr) \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Does this help?