Is there a general way of solving the Maxwell equations?
You need to be more precise about exactly what problem you're solving and what the inputs are. But if you're considering the general problem of what electromagnetic fields are produced by a given configuration of electric charge and current over spacetime, then the general solution is given by Jefimenko's equations.
Look at it as an initial-value problem. If you know the electric and magnetic field throughout space at one instant, and the positions and velocities of all charged particles at that instant, then you can numerically evolve the system forward in time. Two of Maxwell’s equations tell you how fast the fields are changing at each point (and thus their new values after a short time interval), and the Lorentz force law tells you how the particles are accelerating (and thus their new positions and velocities).
The other two Maxwell equations are constraints on the initial condition of the fields, taking the charged particles into account. Figuring out an appropriate initial condition for whatever system you are studying is the harder part of the problem.