Exact analytic solutions for the damped Duffing nonlinear oscillator

We prove that the second-order damped nonlinear Duffing oscillator is reduced to an equivalent equation of the normal Abel form of the second kind. Based on a recently developed mathematical methodology for the construction of exact analytic solutions of Abel's equation, exact analytic solutions are obtained for the nonlinear damped Duffing oscillator obeying the initial conditions adapted to the physical problem. To improve the general developed methodology an application concerning the nonlinear Van der Pol free oscillator is briefly discussed. To cite this article: D.E. Panayotounakos et al., C. R. Mecanique 334 (2006).