On Classification Problems in the Theory of Differential Equations: Algebra $+$ Geometry

Pavel Bibikov, Alexander Malakhov

We study geometric and algebraic approaches to classification problems of differential equations. We consider the so-called Lie problem: provide the point classification of ODEs $y''=F(x,y)$. In the first part of the paper we consider the case of smooth right-hand 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 $y''=F(x,y)$. 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 $y''=F(x,y)$ with rational right hand sides and its symmetry group. This group is a subgroup in the Cremona group of birational automorphisms of $\mathbb{C}^2$, 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.