Asian Journal of Mathematics and Computer Research

This study investigates a numerical approach for approximating solutions of fourth-order ordinary differential equations whose known closed-form solutions can be expressed in Liouvillian-type functions. The work focuses on converting selected fourth-order equations into equivalent systems of first-order ordinary differential equations and solving the resulting systems by the classical fourth-order Runge-Kutta method. The numerical approximations are then compared with known analytical solutions in order to assess the accuracy of the computed values. Two illustrative examples are considered. In the first example, a polynomial solution is approximated, and polynomial fitting is used to examine whether the numerical data can recover the expected quadratic form. In the second example, an equation with exponential and trigonometric components is considered, and the numerical values are used to guide the reconstruction of a solution form involving exponential and sinusoidal terms. The results show close agreement between the Runge-Kutta approximations and the corresponding analytical solutions for the selected test problems and step sizes. The study suggests that numerical sampling, combined with appropriate model fitting, can provide useful guidance in identifying candidate closed-form structures for some fourth-order ordinary differential equations. However, the approach depends on the suitability of the assumed fitting model and on the quality of the numerical data. Therefore, the method should be regarded as a computational aid for selected cases rather than a general replacement for symbolic methods such as differential Galois analysis.