Flame Approximation

I found a really cool example of a differential equation while reading the Matlab tutorial on ODE Solving.

"If you light a match, the ball of flame grows rapidly unitl it reaches a critical size.  Then it remains at that size because the amount of oxygen being consumed by the combustion in the interior of the ball balances the amount available through the surface the simple model is

$\dot{y} = y^2 - y^3 $
$y(0)=\sigma$
$0 \leq t \leq {2/\sigma} $

where $y(t)$ represents the radius of the flame ball, and $\sigma$ is the initial radius."

The analytical solution is

$y(t)=\frac{1}{W(\alpha e^{\alpha -t})+1}$

where $\alpha=1/\sigma -1$ and $W(Z)$ is the Lambert W function, $W(z)e^{W(z)}=z$.

I think it is really amazing how a solution to this problem was constructed by simply relating the change in the radius with time to a surface area term ($y^2$) and a volume term, ($y^3$).  Sometimes I get blinded by wanting to find the exact solution and looking at the details; but here a simple model replicates the details quite accurately.