= Solution
Put $\Omega=1+\delta$ with $|\delta|\ll1$. Linearizing the <cosmological density parameter flow> gives $d\delta/d\log a=(1+3\omega)\delta+O(\delta^2)$. For constant <equation-of-state parameter>,
$$
\boxed{\delta(a)=\delta(a_i)\left(\frac a{a_i}\right)^{1+3\omega}\quad\text{to linear order}.}
$$
For variable $\omega$, the correct replacement is $\delta(a)=\delta(a_i)\exp[\int_{a_i}^{a}(1+3\omega(\widetilde a))\,d\log\widetilde a]$. The printed power assumes constant $\omega$.
For <pressureless matter>, the departure from flatness grows as $a$; for <radiation in cosmology>, it grows as $a^2$. Hence in a conventional decelerating expanding history, maintaining $|\Omega-1|\ll1$ at a late epoch requires extremely small departures at early times. This is the <flatness problem>, not an assertion that exact flatness is inconsistent: $\Omega=1$ is an exact solution, but it is unstable to small departures in <spatial curvature of an FLRW universe> during these eras. With $\omega<-1/3$ the perturbation instead decays, which supplies the <inflationary solution of the flatness problem>.
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