Integrable Equations with Ermakov-Pinney Nonlinearities and Chiellini Damping

We introduce a special type of dissipative Ermakov–Pinney equations of the form  vζζ+g(v)vζ+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)=λ2v , 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)=Ω02(v-v2)  and show that it leads to an integrable hyperelliptic case.