This applet investigates the ideas behind two of Leonhard Euler's numerical approximation techniques for solving simple linear first-order differential equations

UK Year 13 Further Maths - Numerical Methods (AQA FP3, MEI DE)

Instructions below     AQA Further Pure 3 textbook (from AQA site)

[Applet - Euler Methods]

This applet demonstrates two numerical methods for solving linear first-order differential equations.
Method 1 is a first order method, which is easier to apply but less accurate. The error is approximately proportional to the stepsize. It is often called Euler's Method, after the great 18th century Swiss mathematician Leonhard Euler (pronounced 'oiler').

Method 2 is a second order method, since the error is approximately proportional to the square of the stepsize. Hence the error reduces rapidly as the stepsize gets smaller.
You can choose which of the methods to investigate by clicking one of the circles at the bottom of the screen. The stepsize can be doubled or halved using the buttons at the top.
Play around - there are some fascinating patterns to observe!