Integrable equations with Ermakov–Pinney nonlinearities and Chiellini damping

Stefan C. Mancas and Haret C. Rosu

Applied Mathematics and Computation, 2015, vol. 259, issue C, 1-11

Abstract: 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.

Keywords: Dissipative Ermakov–Pinney equation; Chiellini damping; Reid nonlinearities; Abel equation (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:259:y:2015:i:c:p:1-11

DOI: 10.1016/j.amc.2015.02.037

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