= Solution
For pressureless matter with curvature and a possible constant <cosmological constant>, write the background <Friedmann equation> as
$$
H^2=Aa^{-3}+Ba^{-2}+C,\qquad A=H_0^2\Omega_m,\quad B=-k,\quad C=\Lambda/3.
$$
Differentiating on the expanding branch gives
$$
\dot H=-\frac32Aa^{-3}-Ba^{-2},\qquad
\ddot H=H\left(\frac92Aa^{-3}+2Ba^{-2}\right).
$$
Consequently
$$
\ddot H+2H\dot H-\frac32Aa^{-3}H
=H\left[\left(\frac92-3-\frac32\right)Aa^{-3}+(2-2)Ba^{-2}\right]=0.
$$
This proves that the <Hubble parameter as a solution of the dust growth equation> is valid, including constant-$\Lambda$ and curvature contributions to the background. It does not extend without change to a radiation background or arbitrary dynamical dark energy.
In an <Einstein-de Sitter universe>, $a\propto t^{2/3}$, $H=2/(3t)$ and $4\pi G\bar\rho=2/(3t^2)$. The <linear matter perturbation growth equation> becomes
$$
\ddot D+\frac4{3t}\dot D-\frac2{3t^2}D=0.
$$
Substitute $D=t^p$. The indicial equation is $p(p-1)+4p/3-2/3=0$, with roots $2/3$ and $-1$. Therefore
$$
\boxed{D(t)=C_+t^{2/3}+C_-t^{-1},\qquad D_+\propto a,\quad D_-\propto H\propto a^{-3/2}.}
$$
\b[The Hubble-parameter solution is the decaying mode in the Einstein-de Sitter universe.]
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