On the obstruction to linearizability of 2-order ordinary differential equations

Abstract

In this paper, we investigate the action of pseudogroup of all point transformations on the natural bundle of equations $y'' = u^0(x,y) + u^1(x,y)y' + u^2(x,y)(y')^2 + u^3(x,y)(y')^3$. We calculate the 1-st 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 considered bundle. We prove that this form is a unique obstruction to linearizability of these equations by point transformations.