Analysis And Implementation of Euler Method for Solving Ordinary Differential Equation

The Euler method is a basic numerical algorithm for solving ordinary differential equations (ODEs) that occur in different scientific and engineering disciplines. The paper describes a detailed study and application of the Euler method, with a specific focus on solving first-order ODEs. The Euler method is studied with respect to its algorithmic steps, computational complexity, and drawbacks. The technique approximates the solution of ODEs by breaking down the problem into small steps and iterating through them, providing a straightforward and easy-to-understand method for solving initial value problems. Although it is computationally efficient, the precision of the method relies on the step size, with smaller steps providing better approximation but at greater computational expense. The paper also contrasts the Euler method with other higher-order methods, including Runge-Kutta, and discusses its trade-offs. The application of the Euler method is illustrated by a number of example problems, and the outcome is examined both in terms of accuracy and computational efficiency. Lastly, the paper concludes with some recommendations on how the Euler method should be applied in different situations based on its strengths and limitations