= Solution
In the Newtonian dust calculation, use a comoving spherical shell of physical radius $R(t)=a(t)\chi$. Homogeneity gives its enclosed mass $M=4\pi\rho R^3/3$, and the shell theorem removes the gravitational force of exterior shells. Assume pressureless matter, conserved enclosed mass, no <cosmological constant>, and no shell crossing. Its specific conserved mechanical <energy> is
$$
\mathcal E=\frac12\dot R^2-\frac{GM}{R}
=\chi^2\left(\frac12\dot a^2-\frac{4\pi G}{3}\rho a^2\right).
$$
Homogeneity permits the <energy> per $\chi^2$ to be a constant, written $-kc^2/2$. Dividing by $a^2$ gives
$$
\boxed{H^2+\frac{kc^2}{a^2}=\frac{8\pi G}{3}\rho,\qquad H=\dot a/a.}
$$
This is a Newtonian derivation for dust. For radiation the enclosed gravitating mass is not conserved, so this same mechanical-energy argument cannot simply be reused; general relativity supplies the same Friedmann constraint with the total <energy density> divided by $c^2$, together with pressure-dependent fluid conservation.
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