 Solution

Use the supplied flat radiation-era background equations. Write $\overline{\rho }{R}^4=C$, with $C$ constant. On the expanding branch,Choosing the origin at the big bang givesThe supplied perturbation equation consequently becomesA trial $\Delta =t^p$ gives $p(p-1)+p=1$, so $p=\pm 1$. Thus both independent modes areThe growing mode is ${\Delta }_{\mathrm{grow}}\propto t\propto R^2$, 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.