Tangent Lines, Chained

You'll learnto run Euler's method one row at a time, to size and count and point the steps correctly, and to say which side of the true value the estimate lands on before you know the true value.

Euler's method solves a differential equation the way you would cross a dark room: take the direction you have, move a short distance, ask again, repeat. The whole method is one tangent-line step, and the whole difficulty is remembering to ask again. Two failures account for nearly every lost point. One is arithmetic about the steps themselves — the wrong step size, one step too many or too few, a step taken in the wrong direction. The other is conceptual and hides better, because the arithmetic looks identical: reusing the first slope for the whole journey, which quietly replaces the method with a single tangent line.

8 STEPS · 6 QUICK CHECKS · RE-ASK THE SLOPE AT EVERY NEW POINT · v1

dy/dx = x + y, y(0) = 1, h = 0.5 ⇒ 1.5 then 2.5 ⇒ exact 2e − 2 ≈ 3.437

Before you start

What you're looking at

A slope field, and a chain of straight steps walked across it. Each step starts where the last one ended, and points in whatever direction the equation gives at that point.

The question

How wide is a step, how many of them, which way do they go — and does the direction get re-asked at every new point, or reused from the first one?

Watch for

The horizontal and vertical scales are equal in every panel, so a slope of 2 really does look like a slope of 2, and the polyline's turn between its two steps is a turn you can see.

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