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Isothermalwind

-              v1
+              v2
 Use $p=\frac{\rho RT}{\mu}$ and $\frac{RT}{\mu}=a^2$ become:
 $v\frac{dv}{dr}=-\frac{1}{\rho}\frac{d\rho}{dr}-\frac{1}{\rho}\frac{GM}{r^2}+f'(r,v)+g'(r,v)v\frac{dv}{dr}$ (1)
+$v\frac{dv}{dr}=-\frac{1}{\rho}\frac{d\rho}{dr}-\frac{1}{\rho}\frac{GM}{r^2}+f'(r,v)+g'(r,v)v\frac{dv}{dr}$   (1)
 Where $f'(r,v)=\frac{f(r,v)}{\rho}$ $g'(r,v)=\frac{g(r,v)}{\rho}$
 …
 Differentiate $\dot{M}=4\pi r^2\rho v$ with respect to $r$, get
 $\frac{1}{\rho}\frac{d\rho}{dr}=-\frac{1}{v}\frac{dv}{dr}-\frac{2}{r}$ (2)
+$\frac{1}{\rho}\frac{d\rho}{dr}=-\frac{1}{v}\frac{dv}{dr}-\frac{2}{r}$   (2)
 Substitute (2) into (1)
 $\frac{1}{v}\frac{dv}{dr}=\frac{\frac{2a^2}{r}-\frac{GM}{r^2}+f'(r,v)}{(1-g'(r,v))v^2-a^2}$
+$\frac{1}{v}\frac{dv}{dr}=\frac{\frac{2a^2}{r}-\frac{GM}{r^2}+f'(r,v)}{(1-g'(r,v))v^2-a^2}$   (3)
 Critical point will happen at: