= Solution
During <radiation domination>, the <cosmological continuity equation> gives $\rho_R\propto a^{-4}$. The flat <Friedmann equation> in <conformal time> is $(a'/a)^2=8\pi Ga^2\rho_R/3$. Hence $a'^2$ is constant, and choosing the <Big Bang> at $\tau=0$ yields
$$
\boxed{a\propto\tau,\qquad8\pi G\rho_Ra^2=\frac3{\tau^2}.}
$$
For the <CDM density equation in a matter-radiation universe>, neglect $\rho_C\delta_C$ while retaining $\delta_C$ itself. The <adiabatic initial conditions> give $\delta_R=4\delta_C/3$, so
$$
\delta_C''+\frac1\tau\delta_C'=\frac4{\tau^2}\delta_C.
$$
This <Euler-Cauchy equation> has powers $\tau^p$ with $p(p-1)+p-4=0$, namely $p=2,-2$. Selecting the growing <superhorizon adiabatic CDM mode in radiation domination> gives
$$
\boxed{\delta_C(\tau,k)=A_R(k)\tau^2.}
$$
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