= Solution
The linearized <cosmological continuity equation>, the divergence of the <cosmological Euler equation>, and the <cosmological Poisson equation> give
$$
\boxed{\delta'+\theta=0},
\qquad
\boxed{\theta'+\mathcal H\theta
+\frac32\mathcal H^2\Omega_m\delta=0}.
$$
Eliminating $\theta=-\delta'$ yields the equation for a <linear cosmological density perturbation>:
$$
\boxed{\delta''+\mathcal H\delta'
-\frac32\mathcal H^2\Omega_m\delta=0}.
$$
In an <Einstein-de Sitter universe>, $\Omega_m=1$ and $\mathcal H=2/\tau$, so
$$
\delta''+\frac2\tau\delta'-\frac6{\tau^2}\delta=0.
$$
Substitution of the <power-law ansatz> $\delta\propto\tau^p$ gives $(p-2)(p+3)=0$. Hence
$$
\delta=C_+\tau^2+C_-\tau^{-3}
=D_+a+D_-a^{-3/2}.
$$
The leading one of the <matter-era growing and decaying density modes> is $\delta\propto a\propto\tau^2$, as direct substitution confirms, and its velocity divergence is
$$
\boxed{\theta=-\delta'=-\mathcal H\delta\propto-\tau}.
$$
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