= Solution
Use the supplied flat radiation-era background equations. Write $\overline\rho R^4=C$, with $C$ constant. On the expanding branch,
$$
\dot R^2=\frac{8\pi GC}{3R^2},\qquad
R\,\dot R=\sqrt{\frac{8\pi GC}{3}}.
$$
Choosing the origin at the big bang gives
$$
R\propto t^{1/2},\qquad H=\frac{\dot R}{R}=\frac1{2t},
\qquad
\overline\rho=\frac{3}{32\pi Gt^2}.
$$
The supplied perturbation equation consequently becomes
$$
\ddot\Delta+\frac1t\dot\Delta=\frac{\Delta}{t^2}.
$$
A trial $\Delta=t^p$ gives $p(p-1)+p=1$, so $p=\pm1$. Thus \b[both independent modes are]
$$
\boxed{\Delta(t)=C_+t+C_-t^{-1}.}
$$
The growing mode is \b[$\Delta_{\rm grow}\propto t\propto R^2$], and the other decays as $t^{-1}$. At sufficiently late times any nonzero growing-mode coefficient dominates while <linearization> remains valid.
The calculation follows the stated differential equation, including its coefficient $32\pi G/3$. The name Einstein-de Sitter ordinarily denotes a flat pressureless matter universe, whose growth law is different; here the explicit radiation scaling fixes the intended background. Nor does this scale-free equation describe every subhorizon radiation perturbation: including a pressure-gradient term introduces acoustic behavior. No wavelength-independent claim beyond the supplied model is needed.
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