Numerical Methods for Ordinary Differential Equations: A Comprehensive Study of Euler, Runge-Kutta, and Multistep Methods with Truncation Error Analysis, Stability Theory, and Phase Portrait Visualisation

Ordinary differential equations (ODEs) arise ubiquitously in mathematical models of physical, biological, chemical, and engineering systems, yet only a small class possesses closed-form analytical solutions. This paper presents a systematic, mathematically rigorous study of numerical methods for initial value problems y'=f(t,y), y(t₀)=y₀. Beginning with Euler's method — derived from the forward difference approximation — the theory of local truncation error (LTE) and global truncation error (GTE) is developed, proving Euler's method has first-order global accuracy O(h). The classical fourth-order Runge-Kutta method (RK4) is derived via Taylor series matching, and its fourth-order accuracy O(h⁴) established. Multistep Adams-Bashforth methods are derived using polynomial interpolation of past function values, and their orders proved via backward differences. Absolute stability theory is developed using the test equation y'=λy, and stability regions characterised for each method — showing that explicit methods are conditionally stable while implicit methods (Backward Euler, Crank-Nicolson) are unconditionally stable, essential for stiff equations. Complete numerical computations compare Euler, RK2, and RK4 for y'=y, y(0)=1 at h=0.2, and phase portrait analysis of the damped nonlinear pendulum illustrates qualitative ODE dynamics achievable only numerically.