= Solution
Insert the component-density laws into the <Friedmann equation>, using $\rho_{i,0}=3H_0^2\Omega_{i,0}/(8\pi G)$ and $a_0=1$. Multiplying by $a^2$ gives
$$
\boxed{\dot a^2=H_0^2\left(\frac{\Omega_{r,0}}{a^2}
+\frac{\Omega_{m,0}}a+\Omega_{\Lambda,0}a^2\right)-K.}
$$
The <radiation in cosmology> and <pressureless matter> contributions add; the printed first formula has lost the plus sign between them. In the <Friedmann acceleration equation>, $\rho+3P=2\rho_r+\rho_m-2\rho_\Lambda$, so
$$
\boxed{\ddot a=-H_0^2\left(\frac{\Omega_{r,0}}{a^3}
+\frac{\Omega_{m,0}}{2a^2}-\Omega_{\Lambda,0}a\right).}
$$
Evaluating the first equation today gives $K=H_0^2(\Omega_{r,0}+\Omega_{m,0}+\Omega_{\Lambda,0}-1)$.
Now remove <dark energy> and assume a positive <pressureless matter> or <radiation in cosmology> density, starting on an expanding branch. Let $f(a)=H_0^2(\Omega_{r,0}/a^2+\Omega_{m,0}/a)$. It decreases from infinity to zero, and $\dot a^2=f(a)-K$, while $\ddot a<0$. This immediately distinguishes all three cases of <spatial curvature of an FLRW universe>.
* \b[Closed, $K>0$.] There is a unique maximum size $a_{\max}$ with $f(a_{\max})=K$. Expansion from a <Big Bang> slows to $\dot a=0$ and then becomes contraction, ending in a <Big Crunch>. The turning point is reached in finite time because $f(a)-K$ has a simple zero there. The acceleration remains negative at the maximum, so it is a recollapse rather than a static state.
* \b[Open, $K<0$.] The expanding solution never stops because $\dot a^2=f(a)+|K|>0$. It expands forever, eventually approaching a <curvature-dominated universe> with $\dot a\to\sqrt{|K|}$ and $a(t)\sim\sqrt{|K|}\,t$. Locally its late-time behavior approaches the empty <Milne universe>.
* \b[Spatially flat, $K=0$.] Expansion also continues forever, but $\dot a\to0$ instead of approaching a nonzero constant. With <pressureless matter> present, the late-time solution has $a\propto t^{2/3}$, the <Einstein-de Sitter universe> behavior; a flat solution containing only <radiation in cosmology> has $a\propto t^{1/2}$. With both components, an early <radiation domination> era gives way to <matter domination>.
The early small-$a$ solution has $a\propto t^{1/2}$ if <radiation in cosmology> is present, and $a\propto t^{2/3}$ for <pressureless matter> alone. The figure isolates the effect of curvature in three matter-only examples, rather than comparing models with the same present density parameters.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-310-matter-expansion.png]
{title=Matter-only scale-factor evolution: closed recollapse, flat power-law expansion, and open expansion tending to a constant speed.}
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