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

We provide the necessary and sufficient conditions of Liouvillian integrability for nondegenerate near infinity polynomial Levinson-Smith differential equations. These equations generalize Lienard 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.

Automated summary

We investigate the Liouvillian integrability of nondegenerate polynomial Levinson-Smith differential equations near infinity, providing both necessary and sufficient conditions. These equations generalize Lienard equations and are used for describing self-sustained oscillations. Our results are applicable to arbitrary degrees of the polynomials involved. We discover several new Liouvillian integrable subfamilies. For nondegenerate or algebraically degenerate near infinity polynomial Levinson-Smith equations, an upper bound is derived for the degrees of irreducible Darboux polynomials. We also perform a complete classification of Liouvillian first integrals for the cubic Levinson-Smith equation known as the nondegenerate or algebraically degenerate near infinity Rayleigh-Duffing-van der Pol equation.