= Solution
In the smooth-radiation, subhorizon approximation of part (c), the <cosmological Poisson equation> gives
$$
\delta_m''+\mathcal H\delta_m'\simeq4\pi G a^2\bar\rho_m\delta_m
=\frac32H_0^2\Omega_{m,0}a^{-1}\delta_m=\frac{3C}{2y}\delta_m.
$$
From $y'=y\mathcal H$ and part (b),
$$
y'^2=C(1+y),\qquad y''=\frac C2,\qquad\mathcal H y'=\frac{C(1+y)}y.
$$
The expanding branch has $y'>0$. Applying the <chain rule> yields
$$
\delta_m''=C(1+y)\frac{d^2\delta_m}{dy^2}+\frac C2\frac{d\delta_m}{dy},\qquad
\mathcal H\delta_m'=\frac{C(1+y)}y\frac{d\delta_m}{dy}.
$$
Substitute these expressions and divide by $C(1+y)>0$:
$$
\boxed{\frac{d^2\delta_m}{dy^2}+\frac{2+3y}{2y(1+y)}\frac{d\delta_m}{dy}-\frac3{2y(1+y)}\delta_m=0.}
$$
This is the <Mészáros equation> for $y>0$; it keeps radiation in the expansion rate and matter in the slowly varying self-gravity source.
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