Methods for exact solutions of nonlinear ordinary differential equations

In order to find closed form solutions of nonintegrable nonlinear ordinary differential equations, numerous tricks have been proposed. The goal of this short review is to explain how a theorem of Eremenko on meromorphic solutions of some nonlinear ODEs together with some classical, 19th-century results, can be turned into algorithms (thus avoiding ad hoc assumptions) which provide all (as opposed to some) solutions in a precise class. To illustrate these methods, we present some new such exact solutions, physically relevant.