I face some trouble solving Maxwell's equations inside a cylinder with perfect conductor boundaries (in 3D) ? We work with cylindrical coordinates $(r, \phi, z)$ and we make the assumption that fields have a sinusoidal "$e^{i\omega t}$" time dependence. Note that we have a $\phi$ symmetry. First, and in any coordinates system, by taking the rotational and injecting one equation in the other we reduce Maxwell's equations to the following, $$ \nabla\times\nabla\times E = -\partial_t^2 E = \omega^2 E $$ In vacuum, from the $curl curl$ identity, it leads, $$ \nabla\times\nabla\times E = \nabla(\nabla . E) - \nabla^2 E = - \nabla^2 E $$ Where $- \nabla^2 E$ is the laplacian operator applied to each coordinate.
Now, in cylindrical coordinates, we can only compute the $z-$coordinate since, in this case we get the wave equation, $$ \nabla^2 E_z = \omega^2 E_z $$ For the other coordinates, the change of coordinates introduce other terms such that (for the $\phi-$ coordin. ate)$\frac{E_r}{r^2} - \frac{2}{r^2}\frac{\partial E_\phi}{\partial\phi}$.
Then, a fastidious step consists in performing a separation of variable which leads us quite easily to the solution for every separated variable and also to the Bessel differential equation which brings its solution, the Bessel function.
Together with boundary conditions we can get the solution according to $z$ but what about the other coordinates ?