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On solvability of linear differential equations in finite terms

Askold Khovanskii, Aaron Tronsgard

TL;DR

This work develops an algebraic framework for solvability of linear differential equations over differential fields by introducing special transcendental extensions and admissible extension chains. It shows a weak but robust non-solvability result: if an equation is not solvable by generalized quadratures, then special transcendental extensions do not help, at least in the pure transcendental setting, and it builds this through Liouville-inspired analysis of Riccati reductions, order reduction via operator division, and structure of exponentials of integrals. The approach hinges on the generalized Riccati equation, the order-reduction scheme, and the behavior of exponentials of integrals under field extensions to connect solvability to the existence of certain first-order factors. The results illuminate how classical Liouville techniques extend to abstract differential fields and lay groundwork toward a general, algebraic solvability criterion with potential links to differential Galois theory. Overall, the paper offers an accessible, self-contained pathway that mirrors Liouville's original ideas while formalizing them in the differential-algebraic setting.

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.

On solvability of linear differential equations in finite terms

TL;DR

This work develops an algebraic framework for solvability of linear differential equations over differential fields by introducing special transcendental extensions and admissible extension chains. It shows a weak but robust non-solvability result: if an equation is not solvable by generalized quadratures, then special transcendental extensions do not help, at least in the pure transcendental setting, and it builds this through Liouville-inspired analysis of Riccati reductions, order reduction via operator division, and structure of exponentials of integrals. The approach hinges on the generalized Riccati equation, the order-reduction scheme, and the behavior of exponentials of integrals under field extensions to connect solvability to the existence of certain first-order factors. The results illuminate how classical Liouville techniques extend to abstract differential fields and lay groundwork toward a general, algebraic solvability criterion with potential links to differential Galois theory. Overall, the paper offers an accessible, self-contained pathway that mirrors Liouville's original ideas while formalizing them in the differential-algebraic setting.

Abstract

We consider the problem of solvability of linear differential equations over a differential field~. 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 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 . 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.

Paper Structure

This paper contains 13 sections, 16 equations.

Table of Contents

  1. Introduction
  2. Solvability  over differential fields
  3. An illustrative example
  4. Generalized Riccati equation
  5. Reduction of order
  6. Division with a remainder of differential operators
  7. Procedure of order reduction
  8. Pure transcendental extensions
  9. Adjoining a special transcendental element
  10. Existence of a solution in an admissible extension
  11. Algebraic relations between exponentials of integrals
  12. Transcendental extensions by exponentials of integrals
  13. Criterion of solvability in admissible extensions

Theorems & Definitions (21)

  • proof
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  • proof : Proof of Theorem \ref{['I.2']}
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  • ...and 11 more