 Solution

A thin spherical shell has mass $4\pi r^2\rho dr$ and inward gravitational force $Gm(r)(4\pi r^2\rho dr)/r^2$. The net outward pressure force is $-4\pi r^2{P}^{\prime }(r)dr$. Their balance gives the hydrostatic equilibrium equationSince $m^{\prime }=4\pi r^2\rho$, multiplying by $r^2/\rho$ and differentiating yields $\frac{d}{dr}(r^2{P}^{\prime }/\rho )=-4\pi G{r}^2\rho$.

At the centre, require $m(0)=0$ and finite $P(0)$ and $\rho (0)$. Spherical symmetry gives $P^{\prime }(0)=0$; indeed $m(r)=O(r^3)$ makes the pressure gradient $O(r)$ for regular density. At the surface require $P(R)=P_{\mathrm{ext}}$, usually approximated by zero, because pressure matches the surrounding medium. If the stellar mass is prescribed, also impose $m(R)=M$. An equation of state is needed to close the system. A vanishing surface density is appropriate for some stellar models but is not a universal boundary condition.