Elimination of fast variables

First nonlinear ordinary differential equations for the evolution of a physical system are considered. When they involved a small (or large) parameter the evolution occurs on two time scales. A general scheme is developed for extracting equations for the slow evolution, including higher order corrections. The scheme unifies the multifarious methods that exist in the literature. It leads to a subdivision in three categories, determined by the structure of the equations.Although linear equations are merely a specialization, they can readily be generalized to the case of linear partial differential equations, such as the so amply studied Fokker-Planck equation. These are treated with special attention to the definition of the required projection operator.Throughout, numerous examples and applications are given.