The use of the special semi-groups for solving differential equations

• Authors: Jacques-Arthur Weil
• Venue: ISSAC '94 Proceedings of the international symposium on Symbolic and algebraic computation
• Year: 1994

In general, there is no method for finding closed form first integrals or solutions of ordinary differential equations with non-constant coefficients. Thus, one usually performs heuristics, but this involves fastidious computations. The aim of this paper is to propose strategies that computerize such heuristics to help the analysis.In section 1, we formulate our questions in terms of differential algebra. Then, we are able to derive algebraic constructive criteria for the search for closed form solutions of differential equations of the type s(x,y, …y(n–1)y(n) + t(x,y,…,y(n–1)) = 0 (sections 2 and 3). In particular, we focus on the so-called Special polynomials (or Darboux curves). In Section 4, we show how our tools link the expression of the solutions to that of the first integrals, and how it gives a strategy to compute them. Then, in section 5, we show how these techniques permit to derive algorithmic methods to find solutions of order n–1 for linear differential equations of order n; we specifically detail the second order case.