= Solution
In a charge-free and current-free vacuum, <Maxwell equations> give
$$
\nabla\cdot\mathbf E=0,\quad \nabla\cdot\mathbf B=0,\quad \nabla\times\mathbf E=-\partial_t\mathbf B,\quad \nabla\times\mathbf B=\mu_0\epsilon_0\partial_t\mathbf E.
$$
Take the <curl> of <Faraday's law> and use the <curl of the curl identity> $\nabla\times(\nabla\times\mathbf E)=\nabla(\nabla\cdot\mathbf E)-\nabla^2\mathbf E$. The <divergence> term vanishes by <Gauss's law>, while the time derivative of the magnetic <curl> supplies $-\mu_0\epsilon_0\partial_t^2\mathbf E$. Thus
$$
\boxed{\frac1{c^2}\partial_t^2\mathbf E-\nabla^2\mathbf E=\mathbf0,\quad \nabla\cdot\mathbf E=0,\quad c=(\epsilon_0\mu_0)^{-1/2}.}
$$
This is the transverse <electromagnetic wave equation>.
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