Solving differential equations

TL;DR

This paper develops methods to explicitly solve irreducible differential modules using lower-dimensional modules and finite field extensions, leveraging Lie algebra representations and differential Galois theory.

Contribution

It extends classical results by providing explicit solutions for irreducible differential modules through new algebraic techniques.

Findings

Explicit solutions for irreducible differential modules are constructed.

Connections between differential Galois groups and Lie algebra representations are established.

The work generalizes classical results of G. Fano.

Abstract

The theme of this paper is to `solve' an absolutely irreducible differential module explicitly in terms of modules of lower dimension and finite extensions of the differential field $K$. Representations of semi-simple Lie algebras and differential Galois theory are the main tools. The results extend the classical work of G. Fano.