§7.3 Euler's method

Introduction

Euler's Method

The error in Euler's method

Summary

Euler's method approximates solutions of $\frac{dy}{dt} = f(t,y)$ by repeatedly applying

For the initial value problem $\frac{dy}{dt} = \frac{1}{2}(y+1)$ with $y(0) = 0$, the tangent line at $t = 0$ is

Applying Euler's method with $y(0) = 6$ to $\frac{dy}{dt} = 6y-y^2$ produces

For $\frac{dy}{dt} = 6y-y^2$ with $y(0) = 1$ and $\Delta t = 0.2$, compute the first approximation $y_1$.

Continuing with $y_1 = 2$ at $t = 0.2$, compute the second Euler approximation $y_2$ at $t = 0.4$, still with $\Delta t = 0.2$.

The error of Euler's method at a fixed target time is proportional to $\Delta t$, so halving the step size roughly halves the error.