Euler's Method for O.D.E.'s

The first method we shall study for solving differential equations is called Euler's method, it serves to illustrate the concepts involved in the advanced methods. It has limited use because of the larger error that is accumulated with each successive step. However, it is important to study Euler's method because the remainder term and error analysis is easier to understand. Theorem (Euler's Method) Assume that f(t,y) is continuous and satisfies a Lipschits condition in the variable y, and consider the I. V. P. (initial value problem) with , over the interval . Euler's method uses the formulas , and for as an approximate solution to the differential equation using the discrete set of points . Error analysis for Euler's Method When we obtained the formula for Euler's method, the neglected term for each step has the form . If this was the only error at each step, then at the end of the interval , after steps have been made, the accumulated error would be