I've seen several places on the internet say that the constancy of the speed of light can be proven using Maxwell's equations. All of the derivations of the speed of light from Maxwell's equations I've seen, however, seem to use the general wave equation or some other differential equation solution that already assumes the speed of the wave in question is constant.
So far, I understand how to get to this point using Maxwell's equations. $$ \frac{\partial^2 E}{\partial x^2} = \mu_0 \epsilon_0 \frac{\partial^2 E}{\partial t^2}. $$
Is there some way to derive from this partial differential equation that the speed of propagation is constant using only mathematics (i.e. not using anything else that we know experimentally about light)?
Or if you have another derivation entirely, please share.