There exist several methods for computing exact solutions of algebraic differential equations. Most of the methods, however, do not ensure existence and uniqueness of the solutions and might fail after several steps, or are restricted to linear equations. The authors have presented in previous works a method to overcome this problem for autonomous first order algebraic ordinary differential equations and formal Puiseux series solutions and algebraic solutions. In the first case, all solutions can uniquely be represented by a sufficiently large truncation and in the latter case by its minimal polynomial. The main contribution of this paper is the implementation, in a MAPLE-package named FirstOrderSolve, of the algorithmic ideas presented therein. More precisely, all formal Puiseux series and algebraic solutions, including the generic and singular solutions, are computed and described uniquely. The computation strategy is to reduce the given differential equation to a simpler one by using local parametrizations and the already known degree bounds.
翻译:计算代数差异方程式的精确解决办法有几种方法,但大多数方法不能确保解决办法的存在和独特性,而且可能在若干步骤后失败,或限于线性方程式。作者在以前的工作中提出的一种方法,是解决以下问题的方法:一阶自主的代数普通差异方程式和正式的Puiseux系列解决办法和代数法解决办法。在第一种情况下,所有解决办法都可以由足够大的截肢代表,而在后一种情况下,则由最小的多元数值代表。本文的主要贡献是在一个称为PAMALE-Package Afirst OrderSolve的算法组合中实施其中的算法概念。更确切地说,所有正式的Puseux系列和代数解决方案,包括通用和单一的解决方案,都是经过单独计算和描述的。计算策略是使用本地的准位和已知的度界限将给定的差方程式减少到更简单的方程式。