= Solution
During matter domination, pressureless matter has $\bar P_m=\delta P_m=0$. On subhorizon scales the time derivative of the <Newtonian gauge in cosmology> potential is negligible, so the continuity and Euler equations reduce to
$$
\delta_m'=-\nabla\mathbin\cdot\mathbf v,
\qquad
\mathbf v'+\mathcal H\mathbf v=-\nabla\Phi.
$$
Taking the divergence of the second equation, differentiating the first, and using the <Poisson equation> gives
$$
\delta_m''+\mathcal H\delta_m'
-4\pi Ga^2\bar\rho_m\delta_m=0.
$$
Since $d/d\tau=a,d/dt$, one has
$$
\delta_m'=a\dot\delta_m,
\qquad
\delta_m''=a^2(\ddot\delta_m+H\dot\delta_m),
\qquad
\mathcal H=aH.
$$
Division by $a^2$ yields the standard <linear cosmological density perturbation> equation
$$
\boxed{\ddot\delta_m+2H\dot\delta_m
-4\pi G\bar\rho_m\delta_m=0}.
$$
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