Solving second order equations by extending the PS method
  An extension of the ideas of the Prelle-Singer procedure to second order differential equations is proposed. As in the original PS procedure, this version of our method deals with differential equations of the form y''={M(x,y,y')}/{N(x,y,y')}, where M and N are polynomials with coefficients in the field of complex numbers C. The key to our approach is to focus not on the final solution but on the first-order invariants of the equation. Our method is an attempt to address algorithmically the solution of SOODEs whose first integrals are elementary functions of x, y and y'. 
L.G.S. Duarte, L.A. da Mota and J.E.F.Skea (UERJ)