= Solution
Write $y=a/a_0$. Conservation of <pressureless matter> gives $\rho_m=\rho_{m,0}y^{-3}$. The <Friedmann equation>, retaining spatial curvature, is
$$
H^2=H_0^2\bigl(\Omega_{m,0}y^{-3}+\Omega_{k,0}y^{-2}\bigr),
\qquad \Omega_{m,0}+\Omega_{k,0}=1.
$$
Since $\dot y=yH$ and $\Omega_{m,0}=2q_0$, multiplication by $y^2$ gives
$$
\boxed{\left(\frac{\dot a}{a_0}\right)^2
=\dot y^{\,2}=H_0^2\left(1-2q_0+\frac{2q_0}{y}\right).}
$$
In the curvature convention $H^2=8\pi G\rho_m/3-kc^2/a^2$, this also fixes $-kc^2/(a_0^2H_0^2)=1-2q_0$.
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