Source Code for Solving First-Order Ordinary Differential Equations Using Euler's Method and Runge-Kutta Algorithm

In numerical computation, Euler's method and Runge-Kutta algorithm are commonly employed for solving first-order ordinary differential equations. Both algorithms effectively handle ODE solutions through iterative calculation methods. Euler's method provides a straightforward approach by converting differential equations into difference equations, incrementally approximating the exact solution through linear extrapolation. However, Euler's method exhibits relatively low accuracy, particularly with larger step sizes, due to its first-order error approximation. In contrast, the Runge-Kutta algorithm achieves higher precision by calculating function values and slopes at multiple intermediate points within each step interval, typically implementing a weighted average of derivatives. The fourth-order Runge-Kutta (RK4) method, for instance, employs four slope calculations per step to achieve O(h^4) accuracy. The implementation involves key functions for derivative calculation, step-size management, and iterative solution propagation. While more accurate, Runge-Kutta algorithms require increased computational resources and execution time due to multiple function evaluations per step. Therefore, algorithm selection should consider problem complexity, accuracy requirements, and computational constraints. Below we provide comprehensive source code implementations for both Euler's method and Runge-Kutta algorithm, featuring proper error handling and step-size optimization techniques.