Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Liouvillian Solutions of Third Order Differential Equations

Camilo Sanabria, Thierry Combot

Abstract

Consider a third order linear differential equation $L(f)=0$, where $L\in\mathbb{Q}(z)[\partial_z]$. We design an algorithm computing the Liouvillian solutions of $L(f)=0$. The reducible cases devolve to the classical case of second order operators, and in the irreducible cases, only finitely many differential Galois groups are possible. The differential Galois group is obtained through optimized computations of invariants and semi-invariants, and if solvable, the solutions are returned as pullbacks and gauge transformations of algebraic generalized hypergeometric function ${}_3F_2$. The computation time is practical for reasonable size operators.

Liouvillian Solutions of Third Order Differential Equations

Abstract

Consider a third order linear differential equation , where . We design an algorithm computing the Liouvillian solutions of . The reducible cases devolve to the classical case of second order operators, and in the irreducible cases, only finitely many differential Galois groups are possible. The differential Galois group is obtained through optimized computations of invariants and semi-invariants, and if solvable, the solutions are returned as pullbacks and gauge transformations of algebraic generalized hypergeometric function . The computation time is practical for reasonable size operators.
Paper Structure (12 sections, 4 theorems, 18 equations)

This paper contains 12 sections, 4 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Semi-Invariants
  3. List of Semi-Invariants
  4. Symmetric powers
  5. Computing semi-invariants
  6. Riccati Polynomials and Pullbacks
  7. Primitive and Imprimitive groups
  8. The imprimitive case
  9. The primitive case
  10. The algorithm
  11. Examples
  12. Conclusion

Key Result

Corollary 1

The solvable irreducible algebraic subgroups of $SL_3(\mathbb{C})$ are all finite, therefore an irreducible third order differential operator with Galois group in $SL_3(\mathbb{C})$ has Liouvillian solutions if and only if all its solutions are algebraic.

Theorems & Definitions (10)

  • Definition 1
  • Corollary 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof