{
  "abstract": "Differential equations involving reflection of the argument arise in numerous mathematical models where the present state is coupled with its symmetric counterpart. In this paper, we investigate a challenging reflected equation, expressed as y′(t)=ay(t)+by(−t)+∑k=0mcktk, where the forcing function is an arbitrary polynomial of finite degree. A constructive transformation is developed that converts the nonhomogeneous reflected equation into a homogeneous reflection equation. Explicit closed-form expressions for the transformation coefficients are derived and proved by backward induction. Consequently, the original problem is reduced to a homogeneous reflected model whose exact solution is obtained in power-series form and subsequently expressed through hyperbolic or trigonometric functions according to the sign of a2−b2. To complete the analysis, a reduction-of-order technique is employed to establish an equivalent second-order ordinary differential equation without reflection. This approach enables the derivation of exact solutions in the singular cases a=b and a=−b, where the transformation method is no longer applicable. The obtained results provide a unified analytical methodology for a broad class of first-order reflection equations with polynomial forcing and furnish exact closed-form solutions for both regular and resonant parameter regimes.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Mathematics, College of Science, Majmaah University, Al-Majmaah 11952, Saudi Arabia"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Abdulrahman B. Albidah"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5em9heds4erfw.bib",
  "cited_by": 0,
  "doi": "10.3390/axioms15100708",
  "fulltext_url": "https://www.mdpi.com/2075-1680/15/10/708",
  "id": "5em9heds4erfw",
  "issns": [
    "2075-1680"
  ],
  "landing_url": "https://doi.org/10.3390/axioms15100708",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "lcc",
      "QA1-939"
    ],
    [
      "openalex",
      "2604|Nonlinear Differential Equations Analysis"
    ]
  ],
  "openalex": "W7214157270",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/5em9heds4erfw.ris",
  "sources": [
    {
      "first_seen": "2026-10-31T14:21:21Z",
      "native_id": "01207267146d4386a27d545c2be554e4",
      "source": "doaj",
      "url": "https://doi.org/10.3390/axioms15100708"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2604|Nonlinear Differential Equations Analysis",
      "subfield": "general"
    }
  ],
  "title": "Exact Solution for a Class of First-Order Reflected Equations Involving a Polynomial-Type Forcing Term of Arbitrary Finite Degree",
  "title_text": "Exact Solution for a Class of First-Order Reflected Equations Involving a Polynomial-Type Forcing Term of Arbitrary Finite Degree",
  "type": "peer-reviewed",
  "updated_at": "2026-10-31T14:21:21Z",
  "url": "https://research.draconitic.com/paper/5em9heds4erfw",
  "venue": "Axioms",
  "versions": []
}