Four Different Analytical Approaches for the Exact Solution of a Second-Order Scalar Differential Equation with Reflection

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DOI: https://doi.org/10.28924/2291-8639-24-2026-249
Published: Aug 17, 2026
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Laila F. Seddek, Essam R. El-Zahar, Abdelhalim Ebaid, Four Different Analytical Approaches for the Exact Solution of a Second-Order Scalar Differential Equation with Reflection, Int. J. Anal. Appl., 24 (2026), 249.

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Laila F. Seddek, Essam R. El-Zahar, Abdelhalim Ebaid

Abstract

This paper proposes different analytical approaches for solving a class of second-order scalar differential equations involving reflection of the argument. The first approach transforms the given model to a coupled system for which the series solution is obtained in an exact form. The second approach is based on the symmetry decomposition method which derives an equivalent system of two separate equations. Such two separate equations are then solved exactly utilizing well-known standard methods. The third approach converts the problem into a 4th-order ordinary differential equation, allowing the derivation of the same exact solution. The fourth approach applies a direct series method with easier computations of the series coefficients in comparison to the second approach. The obtained solution is expressed in terms of entire functions allowing the appearance of periodicity under a specific constraint of the involved parameters. The results provide a unified procedure for treating reflection-type scalar differential equations.

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