Comperative Solutions to Ordinary Differential Equations Using Euler's and Runge - Kutta Methods

Differential Equations are among the most important Mathematical tools used in creating models in the science, engineering, economics, mathematics, physics, aeronautics, astronomy, dynamics, biology, chemistry, medicine, environmental sciences, social sciences, banking and many other areas. A differential equation that has only one independent variable is called an Ordinary Differential Equation (ODE), and all derivatives in it are taken with respect to that variable. Most often, the variable is time, t; although, I will use x in this paper as the independent variable. The differential equation where the unknown function depends on two or more variables is referred to as Partial Differential Equations (PDE). Ordinary differential equations can be solved by a variety of methods, analytical and numerical. Although there are many analytic methods for finding the solution of differential equations, there exist quite a number of differential equations that cannot be solved analytically. This means that the solution cannot be expressed as the sum of a finite number of elementary functions (polynomials, exponentials, trigonometric, and hyperbolic functions). For simple differential equations, it is possible to find closed form solutions. But many differential equations arising in applications are so complicated that it is sometimes impractical to have solution formulas; or at least if a solution formula is available, it may involve integrals that can be calculated only by using a numerical quadrature formula. In either case, numerical methods provide a powerful alternative tool for solving the differential equations under the prescribed initial condition or conditions. In this paper, I present the basic and commonly used numerical and analytical methods of solving ordinary differential equations ODE. This study presents a comparative analysis on Euler’s and Runge-kutta methods for solving ordinary differential equations (ODEs). The accuracy, stability and computational efficiency of these numerical methods are evaluated and compared. The results show that Runge-kutta methods provide higher accuracy and stability, while Euler’s method offers faster computation. The study highlights the trade-offs between accuracy and computation cost, providing guidelines for seeking the most suitable method for specific ODE problems. The findings contribute to the understanding of numerical methods for ODEs, informing the choice of method for various applications in science and engineering.