Solutions of the Duffing and Painlevé-Gambier Equations by Generalized Sundman Transformation

A new approach using the generalized Sundman transformation to solve explicitly and exactly in a straightforward manner the cubic elliptic Duffing equation is proposed in this study. The method has the advantage to closely relate this equation to the linear harmonic oscillator equation and to be applied to solve other nonlinear differential equations. As a result, explicit and exact general periodic solutions to some Painlevé-Gambier type equations have been established and in particular, it is shown that a reduced Painlevé-Gambier XII equation can exhibit trigonometric solutions, but with a shift factor.