A Note on the Integrability of a Class of Nonlinear Ordinary Differential Equations

Abstract

We study the integrability properties of the hierarchy of a class of nonlinear ordinary differential equations and point out some of the properties of these equations and their connection to the Ermakov-Pinney equation.

Solving Nonlinear Second-Order ODEs via the Eisenhart Lift and Linearization

“…This transition from group to local algebraic representation allowed for the study of invariance properties under these transformations, leading to the linearization of all considered equations and/or functions. Lie's method is a systematic approach designed for the examination of nonlinear systems, which is why it has found extensive application in different subjects within applied mathematics [4][5][6][7][8][9][10][11][12][13][14][15][16][17].…”

Lie symmetries and similarity solutions for a family of 1+1 fifth-order partial differential equations

“…The main novelty of Noether's theorem is the direct relation of the infinitesimal transformations which leave invariant the Action Integral with conservation law for the equation of motions. Consequently, Lie's theory is essential for various physical theories, from classical mechanics, quantum mechanics, fluid theory and many other [5,6,7,8,9,10,11,12,13,14]. Additionally, because the analysis of the Lie symmetries is a systematic way for the determination of exact and analytic solutions of nonlinear differential equations, it plays an important role in all theories of applied mathematics, we refer the reader in [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32] and references therein.…”