New conservation forms and Lie algebras of Ermakov-Pinney equation

Özlem Orhan; Teoman Özer

doi:10.3934/dcdss.2018046

Article Contents

2018, Volume 11, Issue 4: 735-746. Doi: 10.3934/dcdss.2018046

This issue Previous Article Characterization of partial Hamiltonian operators and related first integrals Next Article Differential invariants of a generalized variable-coefficient Gardner equation

New conservation forms and Lie algebras of Ermakov-Pinney equation

Özlem Orhan^1, and
Teoman Özer^2, ,

1.
Istanbul Technical University, Faculty of Science and Letters, Department of Mathematical Engineering, 34469 Maslak, Istanbul, Turkey
2.
Istanbul Technical University, Faculty of Civil Engineering, Division of Mechanics, 34469 Maslak, Istanbul, Turkey

* Corresponding author: Teoman Özer
* Corresponding author: Teoman Özer

Received: November 30, 2016

Revised: April 30, 2017

Published: August 2018

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Abstract

In this study, we investigate first integrals and exact solutions of the Ermakov-Pinney equation. Firstly, the Lagrangian for the equation is constructed and then the determining equations are obtained based on the Lagrangian approach. Noether symmetry classification is implemented, the first integrals, conservation laws are obtained and classified. This classification includes Noether symmetries and first integrals with respect to different choices of external potential function. Furthermore, the time independent integrals and analytical solutions are obtained by using the modified Prelle-Singer procedure as a different approach. Additionally, for the investigation of conservation laws of the equation, we present the mathematical connections between the λ-symmetries, Lie point symmetries and the modified Prelle-Singer procedure. Finally, new Lagrangian and Hamiltonian forms of the equation are determined.

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Mathematics Subject Classification: 22E60, 34A05, 70B05.

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