Solving ordinary differential equations using multistep Leapfrog method

First-order ordinary differential equations (ODEs) are widely used to model various real-world phenomena in science, engineering and applied mathematics. While theoretical methods provide exact solutions for certain classes of ODEs, many practical problems require numerical methods due to the complexity of the equations involved. This study aims to solve ten different first-order ODE problems theoretically and numerically using Euler's method, the second-order Taylor series method, the fourth­order Runge-Kutta (RK4) method and the Leapfrog multistep method. The efficiency of the Leapfrog method as compared to other numerical methods is further analysed based on accuracy and CPU time using Maple code with different step sizes. Performance profiles are constructed using SigmaPlot 15.0 to determine the best numerical method. The results indicate that Euler's method consistently demonstrates the best computational efficiency by producing the lowest CPU time, whereas RK4 provides the best accuracy with the lowest percentage of relative errors. The result also shows that the Taylor-Leapfrog and RK4-Leapfrog methods offer a balanced performance between CPU time and accuracy.