= Solution
During <matter domination>, the spatially flat background has $a\propto\tau^2$ and $\mathcal H=2/\tau$. Let $\theta_m=\nabla\cdot\mathbf v_m$ denote the scalar velocity divergence. Linear <stress-energy conservation> in <synchronous gauge> gives
$$
\delta_m'=-\theta_m-\frac12h',\qquad \theta_m'+\mathcal H\theta_m=0.
$$
The first equation follows from the perturbed volume element $a^3(1+h/2)$; the second states that peculiar dust velocity redshifts as $a^{-1}$. Differentiating the first and substituting the trace of the <Einstein field equations>, with $w_m=0$ and $\Omega_m=1$, yields
$$
\delta_m''+\mathcal H\delta_m'-\frac32\mathcal H^2\delta_m
=-(\theta_m'+\mathcal H\theta_m)=0.
$$
Therefore
$$
\delta_m''+\frac2\tau\delta_m'-\frac6{\tau^2}\delta_m=0.
$$
Trying $\delta_m\propto\tau^p$ gives $p(p-1)+2p-6=(p-2)(p+3)=0$. The complete solution of this density equation is
$$
\boxed{\delta_m(\mathbf x,\tau)=A(\mathbf x)\tau^2+B(\mathbf x)\tau^{-3}.}
$$
These are the <matter-era growing and decaying density modes>, proportional to $a$ and $a^{-3/2}$. In a dust-comoving <synchronous gauge>, $\theta_m=0$ and $h'=-2\delta_m'$. That rest-frame choice distinguishes physical mode coefficients from changes of density induced by <residual synchronous-gauge freedom>.
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