 Solution

For pressureless matter, ${\Delta }_M=\delta -3\Psi$ obeys ${\Delta }_M^{\prime }=-\theta$. The Euler equation givesThe constraint is nowConsequentlyFor a matter-dominated flat background, $a\propto {\eta }^2$ and $\mathcal{H}=2/\eta$. On small scales $\mathcal{G}\ll 1$ but $\mathcal{G}{k}^2\to 9{\mathcal{H}}^2/2$: the last term must not be dropped with $\mathcal{G}$. The equation becomesSince $\Psi$ is small relative to the growing density term on these scales, this recovers $\boxed{{\delta }_{\mathrm{grow}}\propto {\eta }^2\propto a}$.

There is a printed error in the requested super-horizon decaying mode. At $k\ll \mathcal{H}$, $\mathcal{G}\to 1$ and the leading equation is ${\Delta }_M^{\prime \prime }+4{\Delta }_M^{\prime }/\eta =0$, henceThe printed $\alpha +\beta /\eta$ fails its own equation: for ${\Delta }_M=1/\eta$, the left side is $-2/{\eta }^3\ne 0$ at leading order. The $1/\eta$ decaying solution would instead result from using the radiation-era background $\mathcal{H}=1/\eta$ in this limiting damping term. Finite-$k$ corrections restore terms of relative order $(k\eta {)}^2$ and do not repair the wrong leading power.