= Solution
Take the divergence of the linear Euler equation, use $\delta P=c_s^2m\bar n\delta$, and differentiate the continuity equation. Eliminating $\nabla\cdot\delta\mathbf v$ gives
$$
\ddot\delta+2H\dot\delta
-\frac{c_s^2}{a^2}\nabla^2\delta
-\frac1{a^2}\nabla^2\delta\Phi=0.
$$
The perturbed <Poisson equation> is
$$
\nabla^2\delta\Phi=4\pi Gm\bar n a^2\delta.
$$
Thus
$$
\boxed{
\ddot\delta+2H\dot\delta
-c_s^2\frac1{a^2}(\nabla^2+k_J^2)\delta=0,
\qquad
k_J^2=\frac{4\pi Gm\bar n a^2}{c_s^2}}.
$$
Here $k_J$ is the comoving <Jeans wavenumber>.
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