Exact solution of the rate equations for the three energy level laser

The system of two nonlinear differential equations for a three-level laser is solved exactly. It relates the number of photons
q(t)
in the laser cavity and the population inversion
n(t)
. The functions
n(t)
and
q(t)
are expanded in a power series and two recurrence relations are obtained. These are solved and the solutions are in excellent agreement with experimental results.