Differential equations: Euler's method: watch it solved step by step

Follow the slopes of dy/dx = f(x, y) step by step from a starting point, and see how close the estimate is. Level: University. Type any problem of this kind and the Lab works it out and animates every move, with a plain-words explanation of why.

Worked example: dy/dx = 2x, y(0) = 0, h = 0.25, to x = 1

1. Understanding the problem. A differential equation gives the slope of a curve at every point. We know where the curve starts, so we can follow the slopes and trace it out in small steps.
2. Euler's idea. At each point the slope tells us which way the curve is heading. Take one small step h along that tangent: the height changes by about slope × h.
3. The first step. Work out the slope at the start, then step along it.
4. Repeat for every step. Each row uses the end of the row above as its new start.
5. How good is the estimate?. Euler's method follows tangents, so it drifts off a curving solution. Comparing with a much finer method (Runge-Kutta, 20 times more steps) shows the error.
6. The answer. Halving the step size roughly halves the error: smaller steps trade more work for more accuracy.

Answer: y(1) ≈ 0.75 by Euler (a finer method gives ≈ 1)

Practise differential equations: euler's method with problems the Lab marks for you.