Symmetries of nonlinear ordinary differential equations: The modified Emden equation as a case study

Abstract

Abstract. Lie symmetry analysis is one of the powerful tools to analyze nonlinear ordinary differential equations. We review the effectiveness of this method in terms of various symmetries. We present the method of deriving Lie point symmetries, contact symmetries, hidden symmetries, nonlocal symmetries, λ-symmetries, adjoint symmetries and telescopic vector fields of a second-order ordinary differential equation. We also illustrate the algorithm involved in each method by considering a nonlinear oscillator equation as an example. The connections between (i) symmetries and integrating factors and (ii) symmetries and integrals are also discussed and illustrated through the same example. The interconnections between some of the above symmetries, that is (i) Lie point symmetries and λ-symmetries and (ii) exponential nonlocal symmetries and λ-symmetries are also discussed. The order reduction procedure is invoked to derive the general solution of the second-order equation.

Some Remarks on the Solution of Linearisable Second-Order Ordinary Differential Equations via Point Transformations

“…e ODE y ′′ + 3yy ′ + y 3 � 0, (32) called the modified Emdem equation, which arises in a variety of contexts [1], admits the maximal 8 symmetries. Among the admitted symmetries are two noncommuting symmetry generators…”

“…Analytical solutions of many such equations are often hard to find, which is why a whole range of methods have been proposed for investigating different types of nonlinear ODEs. ese methods include Painlevé singularity analysis, Lie symmetry analysis, Darboux method, and the Jacobi last multiplier method (see [1] and the references therein). e Lie symmetry method, which is based on the invariance of a differential equation under a continuous group of point transformations, is widely used.…”

On Lie Symmetry Analysis of Certain Coupled Fractional Ordinary Differential Equations

“…Among those methods, the Lie symmetry analysis method is an algorithmic approach that provides an efficient tool to construct an exact solution of FDEs in a systematic way. Initially, this method was proposed by Norwegian mathematician Sophus Lie during the 19th century and was further developed by Ovsianikov [49] and others [5,10,27,31,34,38,[46][47][48]56,62]. The Lie symmetry analysis method is to find continuous transformations of one or more parameters leaving the differential equation invariant in the new coordinate system wherein the resulting differential equation is easier to solve.…”

Integrability of systems of ordinary differential equations via Lie point symmetries

“…Invariance of ordinary differential equations (ODEs) under one‐parameter Lie groups of transformations has proved to be successful in reduction of order, linearization, classification into equivalent classes, and finding new solutions of the ODEs 1–16 . These successes lead to different integration techniques in order to find complete solution and reduction of the ODEs.…”