Solving Second-Order Differential Equations Using 4th-Order Runge-Kutta Method

In this study, we employ MATLAB programming to solve second-order differential equations using the 4th-order Runge-Kutta numerical method. This widely-used numerical computation approach is applicable to various practical problems across physics, engineering, economics, and other domains involving differential equations. We provide a comprehensive explanation of the method's underlying principles, including the mathematical transformation where second-order equations are converted into systems of first-order equations for numerical processing. The implementation involves MATLAB's vector operations and function handles to create a reusable solver function that calculates four intermediate slopes (k1-k4) at each iteration. Through programmed implementation, we demonstrate step-size control considerations and error analysis techniques. The results are thoroughly analyzed and discussed, with visualization of numerical solutions using MATLAB's plotting capabilities. Furthermore, we conduct comparative analysis of different numerical methods, highlighting the trade-offs between computational accuracy, stability, and efficiency to enhance understanding and application of numerical methods in scientific computing.