Starting from Eq. (4.9) expressed in Lagrangian coordinates, examine small fluctuations about hydrostatic equilibrium by adding to the pressure and radial coordinates (which are functions of mass $m$ in the Lagrangian description) a small time-dependent deviation,
$$
r(m, t)=r_0(m)(1+\delta r(m, t)) \quad P(m, t)=P_0(m)(1+\delta P(m, t)) \text {, }
$$
inserting this into the previous equation, and linearizing the resulting modified equations by expanding small quantities and retaining only linear terms. Show that the deviation in pressure $\delta P$ obeys the differential equation
$$
\frac{\partial(\delta P)}{\partial m}=\frac{1}{4 \pi r_0 P_0}\left(\frac{G m}{r_0^3}(\delta P+4 \delta r)-\frac{\partial^2(\delta r)}{\partial t^2}\right),
$$
where subscript zeros denote the equilibrium values of $r$ and $P$ for a given enclosed mass $m$.