一阶自治AODE的Puixix级数与代数解--一个MAPLE程序包
Puiseux Series and Algebraic Solutions of First Order Autonomous AODEs -- A MAPLE Package
摘要：
存在几种计算代数微分方程精确解的方法。然而，大多数方法不能保证解的存在性和唯一性，并且可能在几个步骤之后失败，或者仅限于线性方程。作者在以前的作品中提出了一种方法，以克服这个问题的自治一阶代数常微分方程和正式Puixix系列的解决方案和代数解决方案。在第一种情况下，所有的解决方案可以唯一地表示为一个足够大的截断，并在后者的情况下，其最小多项式。本文的主要贡献是实现，在一个MAPLE包名为FirstOrderSolve，其中提出的算法思想。更确切地说，所有正式的Puiironx系列和代数解决方案，包括通用和奇异的解决方案，计算和描述唯一。计算策略是利用局部参数化和已知的度界将给定的微分方程简化为一个更简单的微分方程。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
