On an integrable family of oscillators with linear and quadratic damping

Abstract

In this work we study integrability of a family of nonlinear oscillators with linear and quadratic damping. Equations from this family often appear in various applications in physics, mechanics and biology. We demonstrate that certain nonlocal transformations preserve integrating factors for the considered family of equations and provide an explicit expression that connects integrating factors of two nonlocally related oscillators. We apply these results and construct an integrable family of oscillators with linear and quadratic damping, which is connected to an equation from the Painlevé–Gambier classification. We show that members of this family possess an integrating factor, a first integral in terms of the hypergeometric function and a pair of invariant curves. In order to explicitly illustrate our results, we construct new integrable examples of two biologically and physically relevant dynamical systems.