Exact Solutions and Integrability of the Duffing–Van der Pol Equation

The force-free Duffing–Van der Pol oscillator is considered. The truncated expansions for finding the solutions are used to look for exact solutions of this nonlinear ordinary differential equation. Conditions on parameter values of the equation are found to have the linearization of the Duffing–Van der Pol equation. The Painlevé test for this equation is used to study the integrability of the model. Exact solutions of this differential equation are found. In the special case the approach is simplified to demonstrate that some well-known methods can be used for finding exact solutions of nonlinear differential equations. The first integral of the Duffing–Van der Pol equation is found and the general solution of the equation is given in the special case for parameters of the equation. We also demonstrate the efficiency of the method for finding the first integral and the general solution for one of nonlinear second-order ordinary differential equations.