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arXiv · 2609.22962

Exactness as an Explanatory and Computational Target: A Reduction Architecture for First-Order ODEs, with a Higher-Order Outlook

Gabriel Ben-Simon
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Abstract

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.

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Gabriel Ben-Simon. 2026-09-19. Exactness as an Explanatory and Computational Target: A Reduction Architecture for First-Order ODEs, with a Higher-Order Outlook. https://arxiv.org/abs/2609.22962

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