Euler's Method for Differential Equations - Expii

Euler's method applies to first-order ordinary differential equations of the form \(y'=F(x, y)\), given an initial point \((x_0, y_0)\). Specifically, the value of \(y(x)\) can be approximated in \(n\) "steps" as follows: Let \(D=\frac{x-x_0}{n}\). For \(0\le i\le n-1\), let: \(x_i=x_{i-1}+D\) and \(y_i=y_{i-1}+D\times F(x_{i-1}, y_{i-1})\). Then \(x_n=x\) and \(y_n\) will be an approximation of \(y(x)\). Notice that at each step, \(y_i\) is calculated using a tangent line approximation: we are approximately (with step size \(D\)) "tracing along a solution curve in the slope field". Generally speaking, the approximation improves as \(n\) increases and the step size \(D\) decreases.