Expii

# 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.