SOLVING THE FIRST ORDER DIFFERENTIAL EQUATION USING LINEAR MULTISTEP METHOD

First-order ordinary differential equations (ODEs) are essential in modeling dynamic phenomena across science and engineering, yet analytical solutions are often unattainable, necessitating robust numerical methods. This study develops and evaluates linear multistep methods (LMMs) as an efficient alternative to traditional single-step approaches like Euler and Runge-Kutta methods for solving initial value problems of the form y^'=f(x,y), with y(x_0)=y_0. The research derives a continuous LMM using collocation and interpolation techniques, assuming a polynomial approximation of degree 6, interpolated at two points and collocated at five points to determine coefficients. This continuous form is discretized into a 2-point block hybrid method, enabling simultaneous computation of multiple solution values with enhanced stability and self-starting capabilities. Theoretical analysis confirms the method's high order of accuracy, consistency, zero-stability, and absolute stability. Numerical validation on two benchmark problems demonstrates superior performance compared to the fourth-order Runge-Kutta (RK4) method, with LMM errors consistently below 0.001 in the first problem (versus RK4's divergence up to 5.81) and ranging 0.001–0.005 in the second (with better late-stage precision). Graphical error profiles underscore LMM's controlled propagation and reduced sensitivity to step size. The findings highlight LMM's advantages in long-term integration, stiff systems, and computational efficiency, contributing a practical framework for numerical ODE solvers.