On the contact equivalence problem of second order ODEs which are   quadratic with respect to the second order derivative
  In the present paper we establish the necessary and sufficient conditions for two ordinary differential equations of the form $y"{}^2+A(x,y,y') y"+B(x,y,y')=0$ to be equivalent under the action of the pseudogroup of contact transformations. These conditions are formulated in terms of integrals of some one-dimensional distributions. 
Vadim V. Shurygin Jr
math.DG