Analysis of first integrals for some nonlinear differential equations via different approaches

Abstract

This paper begins with first integrals and Lagrangian forms of the Ermakov-Pinney equation. We analyze this equation with the methods which are known as Jacobi last multiplier (JLM) and partial Hamiltonian. The other part of the paper includes a class of the Painleve-Gambier equations and describes the motion of a chain ball drawing with constant force in frictionless surface. The Painleve-Gambier equation is investigated through the following methods: $ \lambda$ -symmetry, Prelle-Singer and partial Hamiltonian. Some of the aforementioned methods have relationships with Lie point symmetries. The first, JLM method, enables us to derive first integrals and Lagrangian forms of ordinary differential equations (ODEs) via Lie point symmetries. The second one is the $ \lambda$ -symmetry method, which is very useful in finding first integrals and integrating factors of ODEs. One way to obtain $ \lambda$ -symmetries is to use Lie point symmetries. Another method, introduced by Naz et al. in 2014 focuses on the partial Hamiltonian systems and is applicable to many problems in various fields, such as applied mathematics, mechanics and economics. Lastly the Prelle-Singer (PS) method has a relation between the $ \lambda$ -symmetry method and null forms, and integrating factors of ODEs can be derived with this connection.