= Solution
During matter domination,
$$
a(t)\propto t^{2/3},
\qquad
\frac{\dot a}{a}=\frac2{3t},
\qquad
\frac{4\pi G\bar\rho_c}{c^2}=\frac2{3t^2}.
$$
The perturbation equation becomes
$$
\ddot\delta_k+\frac4{3t}\dot\delta_k-\frac2{3t^2}\delta_k=0.
$$
Substitution of the power law $\delta_k\propto t^\alpha$ gives
$$
3\alpha^2+\alpha-2=0,
$$
whose roots are $\alpha=2/3$ and $\alpha=-1$. Hence the <cosmic-time matter density modes> give
$$
\boxed{
\delta_k(t)=A(k)\left(\frac{t}{t_{\rm eq}}\right)^{2/3}
+B(k)\left(\frac{t}{t_{\rm eq}}\right)^{-1}.}
$$
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