{
  "abstract": "We examine the nonlinear initial value problem $\\dot{y} = C(1 - y^α)$ modeling polytropic gas permeation. For rational $α$, we derive the exact primitive via cyclotomic partial fraction decomposition, establishing its real trigonometric representation. By synthesizing Ritt's theorem with Rosenlicht's differential algebraic residue criterion, we prove the exact step response $y(t)$ is an elementary function if and only if $α\\in \\{1, 2\\}$. Admitting the Lambert $W$ function extends closed-form solvability strictly to $α\\in \\{-1, 1/2\\}$. Finally, the asymptotic limit $α\\to 0$ converges to a singular dynamics solved by the logarithmic integral. These results constitute a complete classification of all explicit closed-form solutions admitted by this system.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31748",
  "author_total": 1,
  "authors": [
    {
      "name": "Lukáš Mrazík"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dtbf8e4yfxqry.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31748",
  "fulltext_url": null,
  "id": "dtbf8e4yfxqry",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31748",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2608|Advanced Differential Equations and Dynamical Systems"
    ]
  ],
  "openalex": "W7214797186",
  "pdf_url": "https://arxiv.org/pdf/2609.31748",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dtbf8e4yfxqry.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.31748",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31748"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Exact Integration and Invertibility Completeness of the Nonlinear Relaxation Equation",
  "title_text": "Exact Integration and Invertibility Completeness of the Nonlinear Relaxation Equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/dtbf8e4yfxqry",
  "venue": "arXiv",
  "versions": []
}