= Solution
For each Fourier mode, the <cosmological Poisson equation> gives
$$
-k^2\Phi=4\pi Ga^2\bar\rho_m\Delta_m.
$$
Because nonrelativistic matter has $\bar\rho_m\propto a^{-3}$, this implies $\Phi\propto\Delta_m/a$. Write $\Phi=C\Delta_m/a$ and use $\mathcal H=a'/a$. Direct differentiation in the stated potential equation gives
$$
\Delta_m''+\mathcal H\Delta_m'
+(\mathcal H'-\mathcal H^2)\Delta_m=0.
$$
The supplied <Friedmann equation> identity therefore yields
$$
\Delta_m''+\mathcal H\Delta_m'
-4\pi Ga^2\bar\rho_m\Delta_m=0.
$$
Since $d/d\tau=a\,d/dt$, this becomes the <linear growth equation>
$$
\boxed{\ddot\Delta_m+2H\dot\Delta_m
-4\pi G\bar\rho_m\Delta_m=0}.
$$
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