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ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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Applying a Power Transformation to the Orbit of the 2nd Painleve Equation and Solving Differential Equations with Polynomial Right-hand Sides Via the 2nd Painleve Transcendent and in Polynomials

Author(s):

Zilya Nailevna Khakimova

Mozhaisky Military Space Academy
Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Mathematics

vka@mil.ru

Larisa Nikolaevna Timofeeva

Cand. of Sciences (Pedagogy), Associate Professor of the Department of Mathematics
Mozhaisky Military Space Academy

vka@mil.ru

Ajkanush Ashotovna Atojan

Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Mathematics
Mozhaisky Military Space Academy

vka@mil.ru

Abstract:

The 2nd Painleve equation is considered as a representative of the second-order class of ordinary differential equations (ODEs) with polynomial right-hand sides, as well as of the more general second-order class of equations with fractional polynomial right-hand sides. The second Painlevé equation with three terms on the right side has an orbit in the class of fractional polynomial equations with respect to the pseudogroup of the 36th order, and in the absence of the 3rd term – the 60th order. This paper presents a power transformation with an arbitrary parameter that preserves the polynomial or fractional polynomial form of the equations. This power-law transformation is applied to the orbital equations of the 2nd Painlevé equation with three and two terms on the right-hand sides of the equations. Pseudogroups of transformations induced by the above-mentioned pseudogroups of the 36th and 60th orders are constructed. All equations with one-constant arbitrariness corresponding to the vertices of the graphs of induced pseudogroups are found. General solutions of all found equations are obtained through the 2nd Painlevé transcendental or in polynomials. A theorem is presented that allows, using the scaling operation, to find general solutions to all the above equations with arbitrary coefficients.

Keywords

References:

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