Differential Galois Theory and Integration

TL;DR

This paper introduces methods to simplify reducible linear differential systems, revealing classical integrals as solutions and providing insights into algebraic relations among integrals, thereby advancing the understanding of differential Galois theory.

Contribution

It develops new techniques for simplifying reducible linear differential systems and analyzing algebraic relations among integrals, building on previous work in differential Galois theory.

Findings

Methods for reducing differential systems demonstrated on examples

Classical integrals identified as solutions of simplified systems

Insights into algebraic relations between integrals obtained

Abstract

In this paper, we present methods to simplify reducible linear differential systems before solving. Classical integrals appear naturally as solutions of such systems. We will illustrate the methods developed in a previous paper on several examples to reduce the differential system. This will give information on potential algebraic relations between integrals.