Conditional symmetries of nonlinear third-order ordinary differential equations

Aeeman Fatima,Chaudry Masood Khalique,F M Mahomed

doi:10.3934/dcdss.2018040

Abstract

In this work, we take as our base scalar second-order ordinary differential equations (ODEs) which have seven equivalence classes with each class possessing three Lie point symmetries. We show how one can calculate the conditional symmetries of third-order non-linear ODEs subject to root second-order nonlinear ODEs which admit three point symmetries. Moreover, we show when scalar second-order ODEs taken as first integrals or conditional first integrals are inherited as Lie point symmetries and as conditional symmetries of the derived third-order ODE. Furthermore, the derived scalar nonlinear third-order ODEs without substitution are considered for their conditional symmetries subject to root second-order ODEs having three symmetries.

Highlights

Lie classical theory of symmetries for differential equations is an inspiring source for various generalizations aimed at finding ways for obtaining reductions and solutions of differential equations [21]
We investigate that all the Lie point symmetries of the second-order ordinary differential equations (ODEs) considered as first integrals or just derived without substitution of the integral are not necessarily inherited but can be viewed as conditional symmetries of the derived ODE
We have shown by means of an algorithm as to how one can calculates the conditional symmetries of nonlinear third-order ODEs subject to nonlinear second-order ODEs

Summary

Introduction

Lie classical theory of symmetries for differential equations is an inspiring source for various generalizations aimed at finding ways for obtaining reductions and solutions of differential equations [21]. A particular example of this is a variable coefficient linear second-order ODE which has eight Lie point symmetries but is not integrable by quadratures. We investigate that all the Lie point symmetries of the second-order ODEs considered as first integrals or just derived without substitution of the integral are not necessarily inherited but can be viewed as conditional symmetries of the derived ODE.

Results

Conclusion