= Solution
Here primes mean <conformal time> derivatives, $d\tau=dt/a$, and spatial derivatives use comoving coordinates in <Newtonian gauge in cosmology>. Set $\theta_m=\nabla\cdot\mathbf v_m$. The linear <cosmological Euler equation> gives $\theta_m'=-\mathcal H\theta_m-\nabla^2\Phi$, while the continuity equation gives $\theta_m=-\delta_m'+3\Phi'$.
Differentiate the continuity equation and substitute:
$$
\begin{aligned}
\delta_m''&=-\theta_m'+3\Phi''\\
&=\mathcal H\theta_m+\nabla^2\Phi+3\Phi''\\
&=-\mathcal H\delta_m'+\nabla^2\Phi+3\Phi''+3\mathcal H\Phi'.
\end{aligned}
$$
Thus the <linear matter density equation in Newtonian gauge> is
$$
\boxed{\delta_m''+\mathcal H\delta_m'=\nabla^2\Phi+3(\Phi''+\mathcal H\Phi').}
$$
The common potential in these supplied equations assumes that the relevant <scalar anisotropic stress> is neglected. No Fourier-sign convention is needed for the derivation in position space.
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