# Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification

## TL;DR

This paper investigates the solvability of linear differential systems with small coefficients in the Liouvillian sense, extending existing theorems to systems with irregular singular points and providing practical verification methods.

## Contribution

It extends the Ilyashenko-Khovanskii theorem to systems with irregular singular points and small coefficients, linking solvability to explicit Lie algebra conditions.

## Key findings

- Solvability reduces to Lie algebra of coefficient matrices for certain systems.
- Extension of solvability criteria to systems with irregular singular points.
- Practical verification demonstrated using Maple procedures.

## Abstract

We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of solvability of the Lie algebra of the differential Galois group of the system. However, dependence of this Lie algebra on the system coefficients remains unknown. We show that for the particular class of systems with non-resonant irregular singular points that have sufficiently small coefficient matrices, the problem is reduced to that of solvability of the explicit Lie algebra generated by the coefficient matrices. This extends the corresponding Ilyashenko-Khovanskii theorem obtained for linear differential systems with Fuchsian singular points. We also give some examples illustrating the practical verification of the presented criteria of solvability by using general procedures implemented in Maple.