Differential invariants of 2--order ODEs, I

In this paper, we investigate the action of pseudogroup of all point transformations on the natural bundle of equations $y^{\mathrm{\prime }\mathrm{\prime }}=a^3(x,y)(y^{\mathrm{\prime }}{)}^3+a^2(x,y)(y^{\mathrm{\prime }}{)}^2+a^1(x,y)y^{\mathrm{\prime }}+a^0(x,y)$. We construct differential invariants of this action and solve the equivalence problem for some classes of these equations in particular for generic equations.