The Novelty of Shehu Transform on Variable Coefficient Ordinary Differential Equations

Abstract

Variable-coefficient ordinary differential equations (ODEs) are generally more difficult to solve using classical integral transforms, which are primarily suited to constant-coefficient problems. This study develops a generalized analytical method based on the Shehu transform for solving variable-coefficient ODEs. As part of the analytical framework, the Shehu transforms of ty(t), t²y(t), ty′(t), ty″(t), t²y′(t), and t²y″(t) are derived to facilitate the treatment of variable coefficients and derivative-product terms. The proposed method is subsequently applied to several diverse examples, including nonlinear ODEs, to demonstrate its applicability to different classes of variable-coefficient problems. The resulting solutions are consistent with the corresponding exact solutions available in the existing literature, thereby confirming the accuracy and validity of the analytical procedure. These findings demonstrate that the Shehu transform provides an effective and viable framework for solving complex variable-coefficient ODEs. The study contributes generalized operational relations that extend the applicability of the Shehu transform beyond conventional constant-coefficient equations and provide an analytical foundation for addressing broader classes of variable-coefficient differential equations.