We develop a new approach to the study of subgroups in the Cremona group. Namely, we consider the class of differential equations $y''=F(x,y)$ with rational right parts 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.