Solution
= Solution
Differentiate the <linearized cosmological continuity equation>, $\dot\delta=-\theta/a$, and use $\dot a/a=H$:
$$
\ddot\delta=-\frac{\dot\theta}{a}+\frac{H\theta}{a}.
$$
Substitution of part (a), followed by the <Poisson equation> $\nabla^2\Phi=4\pi Ga^2\bar\rho\,\delta$, yields
$$
\boxed{
\ddot\delta+2H\dot\delta
-\frac1{a^2\bar\rho}\nabla^2\delta P
-4\pi G\bar\rho\,\delta=0}.
$$