On Differential Invariants and Classification of Ordinary Differential Equations of the Form y'' = A(x, y)y' + B(x, y)

Abstract

The class of second-order ordinary differential equations y'' = A(x, y)y' + B(x, y) is studied by methods of the geometry of jet spaces and the geometric theory of differential equations. The symmetry group of this class of equations is calculated, and the field of differential invariants of its action on equations is described. These results are used to state and prove a criterion for the local equivalence of two nondegenerate ordinary differential equations of the form y'' = A(x, y)y' + B(x, y), inwhich the coefficients A and B are rational in x and y.