Advanced Engineering Mathematics, 7th Edition

6.1 Euler Methods and Error Analysis

INTRODUCTION

In Section 2.6 we examined one of the simplest numerical methods for approximating solutions of first-order initial-value problems y′ = f(x, y), y(x₀) = y₀. Recall that the backbone of Euler’s method was the formula

(1)

where f is the function obtained from the differential equation y′ = f(x, y). The recursive use of (1) for n = 0, 1, 2, . . . yields the y-coordinates y₁, y₂, y₃, . . . of points on successive “tangent lines” to the solution curve at x₁, x₂, x₃, . . . or x_n = x₀ + nh, where h is a constant and is the size of the step between x_n and x_{n + 1}. The values y₁, y₂, y₃, . . . approximate ...