The Peano-Baker series

TL;DR

This paper provides an elementary, self-contained review of the Peano-Baker series for solving linear ODE systems, including convergence analysis and illustrative examples, serving as a pedagogic introduction.

Contribution

It offers a clear, accessible presentation of the Peano-Baker series with convergence proof and practical examples, filling a gap in pedagogical resources.

Findings

The series converges under specified conditions.

Explicit solutions can be obtained using the series.

Two illustrative examples demonstrate practical application.

Abstract

This note reviews the Peano-Baker series and its use to solve the general linear system of ODEs. The account is elementary and self-contained, and is meant as a pedagogic introduction to this approach, which is well known but usually treated as a folklore result or as a purely formal tool. Here, a simple convergence result is given, and two examples illustrate that the series can be used explicitly as well.