Solution
= Solution
During <matter domination>, $a\propto t^{2/3}$, $H=2/(3t)$, and $4\pi G\bar\rho_m=2/(3t^2)$. Substituting $\Delta_m\propto t^s$ into the <linear growth equation> gives
$$
s(s-1)+\frac43s-\frac23=0,
$$
whose roots are $s=2/3$ and $s=-1$. Therefore
$$
\Delta_m=C_+t^{2/3}+C_-t^{-1}
=C_+a+C_-a^{-3/2}.
$$
The growing one of the <matter-dominated density-perturbation modes> is
$$
\boxed{\Delta_m\propto a}.
$$