发表机构
Afeka Academic College of Engineering（阿费卡工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出以精确代表为统一计算目标组织一阶常微分方程的符号求解，构建双层约化架构，使线性、可分离、齐次、伯努利等方法统一为约化路径，并适应AI与CAS辅助环境。
AI 中文摘要
典型的一阶常微分方程教学大纲常被呈现为一系列具名方法，这可能掩盖了它们之间的统一性。我们提议围绕一个单一的计算目标——精确代表——来组织经典符号一阶大纲的相当一部分内容。对于由一次形式 $\omega$ 表示的正则标量方程，约化问题是在适当的坐标补片上寻找坐标变换 $\Phi$ 和非零因子 $\mu$，使得 $\mu\Phi^*\omega=dG$，从而解曲线为水平集 $G=C$。这产生了一个两层架构。结构层记录目标、变换和定义域限制；操作层执行替换、规范化和积分。线性方程和可分离方程源于最简单的一变量积分因子，而齐次、伯努利及相关类别则成为通向同一目标的约化路径。在人工智能和计算机代数系统辅助的环境中，这使重点转向识别、论证、审计和解释。扩展版本还发展了逆向构造、结构实验实验室、约化路径比较，以及通过首次积分、算子分解和结构规范化的高阶展望。
英文摘要
A typical first-order ODE syllabus is often presented as a sequence of named methods, which can obscure the unity among them. We propose organizing a substantial part of the classical symbolic first-order syllabus around a single computational target: an exact representative. For a regular scalar equation represented by a one-form $ω$, the reduction problem is to find, on an appropriate coordinate patch, a coordinate change $Φ$ and a nonvanishing factor $μ$ such that $μΦ^*ω=dG$, so that solution curves are level sets $G=C$. This yields a two-floor architecture. The structural floor records the target, transformations, and domain restrictions; the operational floor executes substitutions, normalizations, and integrations. Linear and separable equations arise from the simplest one-variable integrating factors, while homogeneous, Bernoulli, and related classes become reduction paths to the same target. In an AI- and CAS-assisted environment, this shifts emphasis toward recognition, justification, auditing, and interpretation. The extended version also develops reverse construction, a structural exercise laboratory, comparisons of reduction paths, and a higher-order outlook through first integrals, operator factorization, and structural normalization.
Comments19 pages, 6 figures
