Solving Second Order Linear Differential Equations via Algebraic Invariant Curves

The idea to find first integrals for polynomial vector fields via algebraic invariant curves can be traced back to Darboux in the 19th century. In 1983, this approach was further developed by Prelle and Singer to become a semi-decision algorithm for finding elementary first integrals. In this paper, we describe how to extend the Prelle-Singer method to deal with second orde..