Lie Theory of Differential Equations and Computer Algebra

Abstract

Introduction The aim of this contribution is to show the possibilities for solving ordinary differential equations with algorithmic methods using Sophus Lie's ideas and computer means. Our material is related especially to Lie's work on transformations and differential equations---essential ideas are already contained in his first paper on transformation groups [5]---and to his article on differential invariants [6]. Very good modern surveys on such questions as are discussed here and on related problems are found in [8,9]. Lie's first intentions were to create a theory for solving differential equations with means of group theory in analogy with the Galois theory for algebraic equations. With respect to typical elements of Galois theory---fields, groups, automorphisms and relations betweeen them---this concept is realized today in the so-called Picard-Vessiot theory for linear ordinary differential equations.