(a) By considering a spherically symmetric star in hydrostatic equilibrium derive the pressure support equation

$$\frac{d P}{d r}=-\frac{G M(r) \rho}{r^{2}},$$

where $r$ is the radial distance from the centre of the star, $M(r)$ is the stellar mass contained inside that radius, and $P(r)$ and $\rho(r)$ are the pressure and density at radius $r$ respectively.

(b) Propose, and briefly justify, boundary conditions for this differential equation, both at the centre of the star $r=0$, and at the stellar surface $r=R$.

Suppose that $P=K \rho^{2}$ for some $K>0$. Show that the density satisfies the linear differential equation

$$\frac{1}{x^{2}} \frac{\partial}{\partial x}\left(x^{2} \frac{\partial \rho}{\partial x}\right)=-\rho$$

where $x=\alpha r$, for some constant $\alpha$, is a rescaled radial coordinate. Find $\alpha$.