= Solution
In the <energy>-conserving blast regime, neglect ambient <pressure>, radiative losses, ejecta mass and gravity compared with the explosion. The cold ambient <ideal gas> has no <sound speed> scale, and its power-law <mass density> has no preferred radius. The point explosion introduces only its <energy> $E$ and elapsed time $t$. This scale-free setting supports a <self-similar blast wave>; extra physics would limit the time interval over which that description applies.
The mass swept up to radius $R$ is
$$
M(R)=4\pi\int_0^R Cr^{2-\beta}\,dr=\frac{4\pi C}{3-\beta}R^{3-\beta}.
$$
It is finite because $\beta<3$. With no <energy> loss, the <kinetic energy> and <internal energy> both scale as $CR^{3-\beta}\dot R^2$. Therefore $R^{3-\beta}\dot R^2$ is constant and integration gives
$$
\boxed{R(t)=A\left(\frac EC\right)^{1/(5-\beta)}t^{2/(5-\beta)}}.
$$
The dimensionless coefficient $A$ depends on the <adiabatic exponent> and ambient exponent through the <similarity profiles>. Indeed, $[E/C]=L^{5-\beta}T^{-2}$, which independently verifies the scaling. The <shock wave> decelerates for $0<\beta<3$, since its time exponent lies between $2/5$ and $1$.
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