Decomposition of ordinary differential equations

Decompositions of linear ordinary differential equations (ode's) into components of lower order have successfully been employed for determining their solutions. Here this approach is generalized to nonlinear ode's. It is not based on the existence of {Lie} symmetries, in that it is a genuine extension of the usual solution algorithms. If an equation allows a {Lie} symmetry, the proposed decompositions are usually more efficient and often lead to simpler expressions for the solution. For the vast majority of equations without a Lie symmetry decomposition is the only available systematic solution procedure. Criteria for the existence of diverse decomposition types and algorithms for applying them are discussed in detail and many examples are given. The collection of Kamke of solved equations, and a tremendeous compilation of random equations are applied as a benchmark test for comparison of various solution procedures. Extensions of these proceedings for more general types of ode's and also partial differential equations are suggested.