A new way to use nonlocal symmetries to determine first integrals of second-order nonlinear ordinary differential equations
Description
Finding first integrals of second-order nonlinear ordinary differential equations (nonlinear 2ODEs) is a very difficult task. In very complicated cases, where we cannot find Darboux polynomials (to construct an integrating factor) or a Lie symmetry (that allows us to simplify the equations), we sometimes can solve the problem by using a nonlocal symmetry. In [1], [2], [3] we developed (and improved) a method (S-function method) that is successful in finding nonlocal Lie symmetries to a large class of nonlinear rational 2ODEs. However, even with the nonlocal symmetry, we still need to solve a 1ODE (which can be very difficult to solve) to find the first integral. In this work we present a novel way of using the nonlocal symmetry to compute the first integral with a very efficient linear procedure.