Solution
= Solution
The Friedmann equation immediately gives
$$
\Omega-1=\frac{kc^2}{\mathcal H^2}.
$$
Using $P=w\rho$ in the <acceleration> equation and eliminating $\rho a^2$ with Friedmann gives
$$
2\mathcal H'+(1+3w)(\mathcal H^2+kc^2)=0.
$$
Differentiate $\Omega-1=kc^2\mathcal H^{-2}$ and substitute this equation:
$$
\Omega'=-2(\Omega-1)\frac{\mathcal H'}{\mathcal H}
=(1+3w)\mathcal H\Omega(\Omega-1),
$$
where $\mathcal H^2+kc^2=\Omega\mathcal H^2$ was used in the last step.