= Solution
Differentiate the <linearized cosmological continuity equation>, remembering the <derivative> of $1/a$:
$$
\ddot\delta_m=\frac{H}{a}\nabla\cdot\mathbf v_m-\frac1a\nabla\cdot\dot{\mathbf v}_m.
$$
Insert the linear <cosmological Euler equation> for the <peculiar velocity>, giving
$$
\ddot\delta_m=\frac{2H}{a}\nabla\cdot\mathbf v_m+\frac1{a^2}\nabla^2\Phi
=-2H\dot\delta_m+4\pi G\bar\rho_m\delta_m.
$$
Thus the <linear matter perturbation growth equation> is
$$
\boxed{\ddot\delta_m+2H\dot\delta_m-4\pi G\bar\rho_m\delta_m=0}.
$$
The factor $2H$ includes one expansion term from continuity and one from momentum dilution. This equation uses sub-Hubble, pressureless linear perturbations and the stated matter-only source for the <gravitational potential>.
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