The Darboux Polynomials and Integrability of Polynomial Levinson–Smith Differential Equations

Abstract

We provide the necessary and sufficient conditions of Liouvillian integrability for nondegenerate near infinity polynomial Levinson–Smith differential equations. These equations generalize Liénard equations and are used to describe self-sustained oscillations. Our results are valid for arbitrary degrees of the polynomials arising in the equations. We find a number of novel Liouvillian integrable subfamilies. We derive an upper bound with respect to one of the variables on the degrees of irreducible Darboux polynomials in the case of nondegenerate or algebraically degenerate near infinity polynomial Levinson–Smith equations. We perform the complete classification of Liouvillian first integrals for the nondegenerate or algebraically degenerate near infinity Rayleigh–Duffing–van der Pol equation that is a cubic Levinson–Smith equation.