On the Obstruction to Linearizability of Second-Order Ordinary Differential Equations

In this paper, we investigate the action of the pseudogroup of all point transformations on the bundle of equations y″=u0(x,y)+u1(x,y)y′+u2(x,y)(y′)2+u3(x,y)(y′)3. We calculate the 1st nontrivial differential invariant of this action. It is a horizontal differential 2-form with values in some algebra, it is defined on the bundle of 2-jets of sections of the bundle under consideration. We prove that this form is a unique obstruction to linearizability of these equations by point transformations.