Galoisian Methods for Testing Irreducibility of Order Two Nonlinear Differential Equations

The aim of this article is to provide a method to prove the irreducibility of non-linear ordinary differential equations by means of the differential Galois group of their variational equations along algebraic solutions. We show that if the dimension of the Galois group of a variational equation is large enough then the equation must be irreducible. We propose a method to compute this dimension via reduced forms. As an application, we reprove the irreducibility of the second and third Painlev\'e equations for special values of their parameter. In the Appendix, we recast the various notions of variational equations found in the literature and prove their equivalences.