Numerical Methods for Solving Ordinary Differential Equations

Ordinary Differential Equations (ODEs) play a central role in modelling physical, biological, engineering, and economic systems. In many real-world applications, exact analytical solutions of ODEs are either difficult or impossible to obtain, necessitating the use of numerical methods. This paper presents a comprehensive study of numerical techniques for solving ordinary differential equations, with emphasis on initial value problems. Classical methods such as Euler’s method, modified Euler methods, Runge–Kutta methods, and multistep methods are discussed with respect to accuracy, stability, convergence, and computational efficiency. The paper also highlights the importance of numerical stability in stiff equations and compares single-step and multi-step approaches. The study aims to provide a clear theoretical foundation along with practical insights into the effective application of numerical methods in solving ODEs.