Publications de l’Institut Mathématique, Nouvelle Série Vol. 103[117] 
On Classification Problems in the Theory of Differential Equations: Algebra $+$ GeometryPavel Bibikov, Alexander MalakhovInstitute of Control Sciences RAS, Moscow, RussiaAbstract: We study geometric and algebraic approaches to classification problems of differential equations. We consider the socalled Lie problem: provide the point classification of ODEs ${y}^{\text{'}\text{'}}=F(x,y)$. In the first part of the paper we consider the case of smooth righthand side $F$. The symmetry group for such equations has infinite dimension, so classical constructions from the theory of differential invariants do not work. Nevertheless, we compute the algebra of differential invariants and obtain a criterion for the local equivalence of two ODEs ${y}^{\text{'}\text{'}}=F(x,y)$. In the second part of the paper we develop a new approach to the study of subgroups in the Cremona group. Namely, we consider class of differential equations ${y}^{\text{'}\text{'}}=F(x,y)$ with rational right hand sides and its symmetry group. This group is a subgroup in the Cremona group of birational automorphisms of ${\u2102}^{2}$, which makes it possible to apply for their study methods of differential invariants and geometric theory of differential equations. Also, using algebraic methods in the theory of differential equations we obtain a global classification for such equations instead of local classifications for such problems provided by Lie, Tresse and others. Keywords: differential equation, jet space, symmetry group, differential invariant, algebraic manifold, polynomial dependence Classification (MSC2000): 34A26, 58A20, 14E07 Full text of the article: (for faster download, first choose a mirror)
Electronic fulltext finalized on: 26 Apr 2018. This page was last modified: 11 Mai 2018.
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