Editing SSCpt2/SolutionStrategies (section)

===Technique 1===
Integrating the Poisson equation once, from the center of the configuration <math>~(r=0)</math> out to some finite radius, <math>~r</math>, that is still inside the configuration, gives,

<div align="center">
<math> 
~\int_0^r d\biggl( r^2 \frac{d \Phi}{dr} \biggr)  = \int_0^r 4\pi G r^2 \rho dr 
</math><br />

<math> 
\Rightarrow ~~~~~ r^2 \frac{d \Phi}{dr} \biggr|_0^r  = GM_r \, .
</math>
</div>

Now, as long as <math>~d\Phi/dr</math> increases less steeply than <math>~r^{-2}</math> as we move toward the center of the configuration &#8212; indeed, we will find that <math>~d\Phi/dr</math> usually goes smoothly to zero at the center &#8212; the term on the left-hand-side of this last expression will go to zero at <math>~r=0</math>.  Hence, this first integration of the Poisson equation gives,

<div align="center">
<math> 
~\frac{d \Phi}{dr}  = \frac{G M_r}{r^2} \, .
</math>
</div>

Substituting this expression into the hydrostatic balance equation gives,

<div align="center">
{{Math/EQ_SShydrostaticBalance01}}
</div>

that is, a single governing integro-differential equation which depends only on the two unknown functions, {{Math/VAR_Pressure01}} and {{Math/VAR_Density01}} .