This paper presents an alternative derivation of an integrating factor for obtaining analytical solutions of second-order ordinary linear differential equations (ODEs) with variable polynomial coefficients, as originally introduced in earlier research. Building on that work, we present a new derivation of the integrating factor based on the concept of exactness in higher-order ordinary differential equations. The resulting integrating factor is obtained in explicit form. To assess the effectiveness of the proposed method, it is applied to two well-known second-order linear differential equations, namely the Bessel equation of order 1/2 and the modified Laguerre equation. The results demonstrate that the method produces analytical solutions for both cases and produces corresponding analytical solutions and graphical illustrations The applicability of the method depends on specific forms of the auxiliary function S(u) that admit closed-form solutions.
Jama, M. Abdirahman, Gathungu, D. Kioi, Onyango, E. Richard & Chepkonga, D. (2026). An Alternative Integrating-Factor Framework for Certain Classes of Second-Order Linear Ordinary Differential Equations. (e742894). Mathematics and Computational Sciences, (), e742894 https://doi.org/10.30511/mcs.2026.2083591.1700
MLA
Jama, M. Abdirahman, Gathungu, D. Kioi, Onyango, E. Richard, & Chepkonga, D. "An Alternative Integrating-Factor Framework for Certain Classes of Second-Order Linear Ordinary Differential Equations" .e742894 , Mathematics and Computational Sciences, , 2026, e742894. doi: 10.30511/mcs.2026.2083591.1700
HARVARD
Jama, M. Abdirahman, Gathungu, D. Kioi, Onyango, E. Richard, & Chepkonga, D. (2026). 'An Alternative Integrating-Factor Framework for Certain Classes of Second-Order Linear Ordinary Differential Equations', Mathematics and Computational Sciences, (), e742894. doi: 10.30511/mcs.2026.2083591.1700
CHICAGO
Jama, M. Abdirahman, Gathungu, D. Kioi, Onyango, E. Richard & Chepkonga, D., "An Alternative Integrating-Factor Framework for Certain Classes of Second-Order Linear Ordinary Differential Equations." Mathematics and Computational Sciences, (2026): e742894, doi: 10.30511/mcs.2026.2083591.1700
VANCOUVER
Jama M. Abdirahman, Gathungu D. Kioi, Onyango E. Richard, Chepkonga D. An Alternative Integrating-Factor Framework for Certain Classes of Second-Order Linear Ordinary Differential Equations. MCS. 2026;():e742894. doi: 10.30511/mcs.2026.2083591.1700
