A spherically-symmetric star obeys the pressure-support equation

$$\frac{d P}{d r}=-\frac{G m \rho}{r^{2}}$$

where $P(r)$ is the pressure at a distance $r$ from the centre, $\rho(r)$ is the density, and $m(r)$ is the mass within a sphere of radius $r$. Show that this implies

$$\frac{d}{d r}\left(\frac{r^{2}}{\rho} \frac{d P}{d r}\right)=-4 \pi G r^{2} \rho$$

Propose and justify appropriate boundary conditions for the pressure $P(r)$ at the centre of the star $(r=0)$ and at its outer edge $r=R$.

Show that the function

$$F(r)=P(r)+\frac{G m^{2}}{8 \pi r^{4}}$$

is a decreasing function of $r$. Deduce that the central pressure $P_{\mathrm{c}} \equiv P(0)$ satisfies

$$P_{\mathrm{c}}>\frac{G M^{2}}{8 \pi R^{4}},$$

where $M \equiv m(R)$ is the mass of the star.