On the solution of nonlinear, second order, autonomous systems

Abstract

The author's previous work on an extension of Abel's identity to nonlinear second order systems is applied to the construction of implicit general solutions to two complicated nonlinear systems. Both systems are nonconservative. One has a quadratic nonlinearity in the response terms and the other has a cubic nonlinearity. The exact general solutions are shown to involve a definite integral with respect to one dependent variable holding the other fixed. The unconventional nature of these integrals does not detract from their potential value. Instead, it shows very clearly why some conventional tools fail to produce results when applied to such systems.