  This work introduces a methodology to solve ordinary differential equations
using the Schur decomposition of the linear representation of the differential
equation. This is done by first transforming the system into an upper
triangular system using the Schur decomposition, and second, by generating the
solution sequentially following the upper triangular structure. In addition,
and when dealing with non-linear perturbed systems, this work proposes a
methodology based on operator theory to find an approximate linear
representation of perturbed non-linear systems. Particularly, we focus on
polynomial differential equations and the use of Legendre polynomials to
represent the solution. Based on these results, a perturbation technique is
also proposed to study these problems. A set of algorithms to automate these
methodologies are included.
