Exact analytic solutions for the damped Duffing nonlinear oscillator (Q869410)

In two previous papers the authors found a closed form solution for the normal form \[ yy'-y=f(x) \] of the Abel equation of the second kind. In the given paper, the authors reduce the damped Duffing oscillator to such an equation, so that it becomes also possible to solve this equation. A complication arises from the fact that the solutions may change its character at some point which must be determined. Furthermore, the explicit determination of the solution requires to solve transcendental equations which is usually only approximately possible.