= Solution
Use <mass density> units for the $\rho$ appearing in the <Friedmann equation>; an actual <energy density> is $\rho c^2$. Conservation, $\dot\rho+3H(\rho+P/c^2)=0$, gives $\boxed{\rho_M=\rho_{M,0}a^{-3},\quad\rho_R=\rho_{R,0}a^{-4}}$ for <pressureless matter> and <radiation in cosmology>, respectively, with $a_0=1$. The first scaling counts dilution of particle number, while the second includes the <cosmological redshift> of each photon's <energy>.
Define the present <cosmological density parameters> by $\Omega_{X,0}=8\pi G\rho_{X,0}/(3H_0^2)$. Then
$$
\frac{H^2}{H_0^2}=\Omega_{M,0}a^{-3}+\Omega_{R,0}a^{-4}+\Omega_{k,0}a^{-2},\qquad
\Omega_{k,0}=-\frac{kc^2}{H_0^2}=1-\Omega_{M,0}-\Omega_{R,0}.
$$
For $k=0$, <conformal time> gives $da/d\eta=a^2H$, so the expanding branch obeys $(da/d\eta)^2=H_0^2(\Omega_{M,0}a+\Omega_{R,0})$. Set $\eta=0$ at the <Big Bang>. Integrating gives
$$
\sqrt{\Omega_{M,0}a+\Omega_{R,0}}=\sqrt{\Omega_{R,0}}+\frac12\Omega_{M,0}H_0\eta,
$$
and hence the <flat matter-radiation conformal solution>
$$
\boxed{a(\eta)=\sqrt{\Omega_{R,0}}H_0\eta+\frac14\Omega_{M,0}(H_0\eta)^2.}
$$
The limiting pure-radiation and pure-matter cases follow continuously.
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