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

An Analog of Variational Symmetry for Odd Order ODEs

Author(s):

Ngi Huan Hoang

postgraduate student
RGPI by A.I.Gerzen

huanvietnam@mail.ru

Abstract:

In this paper we present the inverse problem of Group Analysis Theory: a search of broad classes of 3rd- order equations that admit analogues of variational symmetry, i. e. equations which have first integrals inheriting point symmetry of initial equations. We first summarize the development of Group Analysis Theory, then propose problem statement and explain our main method that facilitates the solution of inverse problem. The paper will introduce results of a search for equivalent groups of three classes of the equations: y’’’=F(x, y), y’’’=F(x, y, y’), y’’’=F(x, y, y’’). We formulate and strictly prove theorems on the existence and structure of the equations of given classes y'''=F(y), y'''=F(y,y'),y'''=F(y)y''+G(y) и y'''=F(y)(y'')^2+G(y)y''+H(y) having first integrals in polynomial forms (linear, square or cubic forms). We also consider the simplest cases for the class y'''=F(y,y',y'')G(y'').

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