Exact equations to design aplanatic sequential optical systems

We present the exact differential equations to design an aplanatic sequential optical system, a system that is free of spherical aberration and linear coma. We get the exact set of equations from the Fermat principle and the Abbe sine condition. We solve the mentioned set of equations by implementing the Runge-Kutta algorithm. We test the solutions using commercial ray-tracing software and confirm the expected behavior of the optical system.