{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T15:41:36Z","timestamp":1767109296875,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,2,19]],"date-time":"2024-02-19T00:00:00Z","timestamp":1708300800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"research support program of Shaqra University, Shaqra, Saudi Arabia"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The existence of the advance parameter in a scalar differential equation prevents the application of the well-known standard methods used for solving classical ordinary differential equations. A simple procedure is introduced in this paper to remove the advance parameter from a special kind of first-order scalar differential equation. The suggested approach transforms the given first-order scalar differential equation to an equivalent second-order ordinary differential equation (ODE) without the advance parameter. Using this method, we are able to construct the exact solution of both the transformed model and the given original model. The exact solution is obtained in a wave form with specified amplitude and phase. Furthermore, several special cases are investigated at certain values\/relationships of the involved parameters. It is shown that the exact solution in the absence of the advance parameter reduces to the corresponding solution in the literature. In addition, it is declared that the current model enjoys various kinds of solutions, such as constant solutions, polynomial solutions, and periodic solutions under certain constraints of the included parameters.<\/jats:p>","DOI":"10.3390\/axioms13020129","type":"journal-article","created":{"date-parts":[[2024,2,19]],"date-time":"2024-02-19T03:18:38Z","timestamp":1708312718000},"page":"129","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach"],"prefix":"10.3390","volume":"13","author":[{"given":"Nada A. M.","family":"Alshomrani","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1122-6297","authenticated-orcid":false,"given":"Abdelhalim","family":"Ebaid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Faten","family":"Aldosari","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Shaqra University, P.O. Box 15572, Shaqra 11961, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mona D.","family":"Aljoufi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1243\/PIME_PROC_1964_179_050_02","article-title":"Third paper: Calculating the behaviour of an overhead catenary system for railway electrification","volume":"179","author":"Andrews","year":"1964","journal-title":"Proc. Inst. Mech. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1093\/comjnl\/13.4.363","article-title":"Numerical method for calculating the dynamic behaviour of a trolley wire overhead contact system for electric railways","volume":"13","author":"Abbott","year":"1970","journal-title":"Comput. 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