= Solution
For a flat matter-dominated universe, the Friedmann solution has $a\propto t^{2/3}$, $H=2/(3t)$ and $\bar\rho=1/(6\pi Gt^2)$. Thus the density equation is
$$
\ddot\delta+\frac4{3t}\dot\delta-\frac2{3t^2}\delta=0.
$$
Trying $t^s$ gives $s(s-1)+4s/3-2/3=0$, with roots $2/3$ and $-1$. Therefore
$$
\boxed{\delta(t)=A t^{2/3}+B t^{-1}=A' a+B'a^{-3/2}}.
$$
The growing mode is proportional to the <scale factor>. When smooth <dark energy> or a <cosmological constant> dominates, the matter source density falls while expansion provides continued damping, suppressing growth. In the late constant-$\Lambda$ limit, $H\to H_\Lambda$ and $\bar\rho\to0$, so $\ddot\delta+2H_\Lambda\dot\delta\simeq0$ and $\delta$ tends to a constant plus a decaying $e^{-2H_\Lambda t}$ term. Matter perturbations therefore cease growing appreciably rather than continuing the matter-era law indefinitely.
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