Third order differential equations with fixed critical points

The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painlevé property are explored for ODEs of order n = 2, . . . , 5. Among the 6 ODEs identifying the Painlevé transcendents only PIII , PV and PV I have nontrivial symmetry algebras and that only for very special values of the parameters. In those cases the transcendents can be expressed in terms of simpler functions, i.e. elementary functions, solutions of linear equations, elliptic functions or Painlevé transcendents occurring at lower order. For higher order or higher degree ODEs that pass the Painlevé test only very partial classifications have been published. We consider many examples that exist in the literature and show how their symmetry groups help to identify those that may define genuinely new transcendents.

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The first integrals and rational solutions of some fourth-order differential equations

Babich

Martynov

2020

Vescì Akademìì navuk Belarusì. Seryâ fizika-matematyčnyh navuk

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The object of this research is fourth-order differential equations. The aim of the research is to study the analytical properties of the solutions of these differential equations. The general form of the considered equations is indicated, and also the choice of the research object is justified. Herein we studied fourth-order differential equations for which sets of resonances with all positive nontrivial resonances are absent. Besides, three of these equations satisfy the conditions of absence in the solutions of moving multivalued singular points. The solutions of the next three equations have movable special points of multivalued character. Moreover, we also investigated the analytical properties of one more fourth-order differential equation of another general form for which it is also possible to construct a two-parameter rational solution as there is a nontrivial negative resonance in the related set of resonances. The first integrals of the equations under study are found and their rational solutions are constructed from negative non-trivial resonances. The resonance method was used in this study. The obtained results can be used in the analytical theory of differential equations.

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Third order differential equations with fixed critical points

Abstract: a b s t r a c tThe singular point analysis of third order ordinary differential equations which are algebraic in y and y 0 is presented. Some new third order ordinary differential equations that pass the Painlevé test as well as the known ones are found.

Cited by 3 publications

References 40 publications

Soliton, rational and special solutions of the Korteweg–de Vries hierarchy

Soliton, rational and special solutions of the Korteweg–de Vries hierarchy

Lie point symmetries and ODEs passing the Painlevé test

The first integrals and rational solutions of some fourth-order differential equations

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