Integrable equations with Ermakov-Pinney nonlinearities and Chiellini damping

Abstract

"We introduce a special type of dissipative Ermakov-Pinney equations of the form v?? + g(v)v? + h(v) = 0, where h(v) = h0(v) + cv?3 and the nonlinear dissipation g(v) is based on the corresponding Chiellini integrable Abel equation. When h0(v) is a linear function, h0(v) = ?2v, general solutions are obtained following the Abel equation route. Based on particular solutions, we also provide general solutions containing a factor with the phase of the Milne type. In addition, the same kinds of general solutions are constructed for the cases of higher-order Reid nonlinearities. The Chiellini dissipative function is actually a dissipation-gain function because it can be negative on some intervals. We also examine the nonlinear case h0(v) = ?20(v ? v2) and show that it leads to an integrable hyperelliptic case."