= Solution
Let $H=\dot a/a$ be the <Hubble parameter>, and normalize the <scale factor> by $a(t_0)=1$. The <cosmological continuity equation> gives $\rho_M=\rho_{M0}a^{-3}$ for <pressureless matter>, whereas the <cosmological constant> has constant energy density $\rho_\Lambda$. The spatially flat <Friedmann equation> therefore becomes
$$
H^2=\frac{8\pi G}{3}(\rho_{M0}a^{-3}+\rho_\Lambda).
$$
The present <Hubble constant> is $H_0=H(t_0)$. In terms of the present <critical density> $\rho_{\mathrm{crit},0}=3H_0^2/(8\pi G)$, the matter <cosmological density parameter> is $\Omega_M=\rho_{M0}/\rho_{\mathrm{crit},0}$. Spatial flatness gives $\rho_\Lambda/\rho_{\mathrm{crit},0}=1-\Omega_M$, so
$$
\boxed{\left(\frac{\dot a}{a}\right)^2=H_0^2\left[\frac{\Omega_M}{a^3}+1-\Omega_M\right].}
$$
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