Reduction of fourth order ordinary differential equations to second and third order Lie linearizable forms

In this paper, the linearization of third-order ordinary differential equations, which are the transformed equations of the quintic nonlinear beam equation, is presented. First of all, a third-order ordinary differential equation can be linearized if its coefficients satisfied the conditions of the linearization theorem. So, the conditions for linearization are investigated. After that, in each case, the linearizing transformation is defined, and finally the linear third-order equation is obtained. Moreover, after calculating the solutions to linear third-order equations and substituting the original variables to the solutions, the exact solutions to the equation of motion are obtained.

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Classification of scalar third order ordinary differential equations linearizable via generalized contact transformations

Dutt

Qadir

2017

Quaestiones Mathematicae

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Whereas Lie had linearized scalar second order ordinary differential equations (ODEs) by point transformations and later Chern had extended to the third order by using contact transformation, till recently no work had been done for higher order or systems of ODEs. Lie had found a unique class defined by the number of infinitesimal symmetry generators but the more general ODEs were not so classified. Recently classifications of higher order and systems of ODEs were provided. In this paper we relate contact symmetries of scalar ODEs with point symmetries of reduced systems. We define new type of transformations that build up this relation and obtain equivalence classes of scalar third order ODEs linearizable via these transformations. Four equivalence classes of such equations are seen to exist.

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Reduction of fourth order ordinary differential equations to second and third order Lie linearizable forms

Cited by 3 publications

References 15 publications

Complex Methods for Lie Symmetry Analysis

Complex Methods for Lie Symmetry Analysis

Conditional linearization of the quintic nonlinear beam equation

Classification of scalar third order ordinary differential equations linearizable via generalized contact transformations

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