On solvability of linear differential equations in finite terms

TL;DR

This paper investigates the conditions under which linear differential equations over a differential field can be solved using finite terms, introducing a new class of field extensions and demonstrating the limitations of generalized quadratures.

Contribution

It introduces a new class of differential field extensions that generalize classical extensions and proves that certain equations cannot be solved by generalized quadratures even with these extensions.

Findings

If an equation cannot be solved by generalized quadratures, no special extension can solve it.

A weaker version of the main result is proven for pure transcendental extensions.

The paper demonstrates the effectiveness of Liouville's approach to solvability in finite terms.

Abstract

We consider the problem of solvability of linear differential equations over a differential field~$K$. We introduce a class of special differential field extensions, which widely generalizes the classical class of extensions of differential fields by integrals and by exponentials of integrals and which has similar properties. We announce the following result: if a linear differential equation over $K$ can not be solved by generalized quadratures, then no special extension can help solve it. In the paper we prove a weaker version of this result in which we consider only pure transcendental extensions of $K$. Our paper is self-contained and understandable for beginners. It demonstrates the power of Liouville's original approach to problems of solvability of equations in finite terms.