Solving second order ordinary differential 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 ′ .