= Solution
Define the present <critical density> and <cosmological density parameter> by
$$
\rho_{\rm crit,0}=\frac{3H_0^2}{8\pi G},\qquad\Omega_{{\rm DE},0}=\frac{\rho_{{\rm DE},0}}{\rho_{\rm crit,0}}.
$$
Spatial flatness with $\Lambda=0$ implies $\Omega_{M,0}=1-\Omega_{{\rm DE},0}$. <Separately conserved cosmological fluids> and <constant-equation-of-state density scaling> give $\rho_M=\rho_{M,0}a^{-3}$ and $\rho_{\rm DE}=\rho_{{\rm DE},0}a^{-3(1+w)}$. Inserting both into the <Friedmann equation> yields
$$
\boxed{H(a)^2=H_0^2\left[(1-\Omega_{{\rm DE},0})a^{-3}+\Omega_{{\rm DE},0}a^{-3(1+w)}\right]=H_0^2\left[\Omega_{{\rm DE},0}a^{-3}(a^{-3w}-1)+a^{-3}\right].}
$$
This is the constant-$w$ case of the <Hubble parameter for matter and dynamical dark energy>.
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