Solving second-order ordinary differential equations by extending the Prelle-Singer method

We propose a method for solving second-order ordinary differential equations (ODEs) which is based on the ideas behind the Prelle-Singer (PS) procedure for first-order ODEs. While the PS procedure treats differential equations (DEs) of the form y' = P(x, y)/Q(x, y), with P and Q polynomials whose coefficients lie in the field of complex numbers C, our method is applicable to DEs of the form y" = P(x, y, y')/Q(x, y, y'). 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 second-order ODEs with solutions in terms of elementary functions.