= Solution
In <vacuum>, <Maxwell's equations> in differential form are
$$
\nabla\cdot\mathbf E=\frac{\rho}{\epsilon_0},\qquad \nabla\cdot\mathbf B=0,\qquad
\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t},\qquad
\nabla\times\mathbf B=\mu_0\mathbf J+\mu_0\epsilon_0\frac{\partial\mathbf E}{\partial t}.
$$
Here $\rho$ is <charge density>, $\mathbf J$ is <electric current density>, $\mathbf E$ is <electric field>, and $\mathbf B$ is <magnetic flux density>. In a source-free region, taking the <curl> of the third equation and using $\nabla\times(\nabla\times\mathbf E)=\nabla(\nabla\cdot\mathbf E)-\nabla^2\mathbf E$ gives the <wave equation>
$$
\nabla^2\mathbf E=\mu_0\epsilon_0\frac{\partial^2\mathbf E}{\partial t^2}.
$$
The same argument gives the <wave equation> for $\mathbf B$. A concrete family satisfying all four <Maxwell's equations> is
$$
\mathbf E=\widehat{\mathbf x}F(z-ct),\qquad \mathbf B=\frac{\widehat{\mathbf y}}{c}F(z-ct),\qquad
\boxed{c=\frac1{\sqrt{\mu_0\epsilon_0}}.}
$$
The function $F$ may be any twice differentiable profile. These are transverse <travelling waves>; reverse the <magnetic field> direction for propagation in the opposite direction. Because the source-free <Maxwell's equations> are homogeneous <linear differential equations>, any finite <linear combination> of such paired <electric fields> and <magnetic fields> is again a solution. Infinite <superpositions> also work whenever the required derivatives converge. A solution of the component <wave equations> alone must still satisfy the divergence and mutual-curl constraints.
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