Slope Fields and Euler's Method Simulation

A differential equation does not hand you a curve; it hands you a rule for how steep the curve must be at every point. Drawn as a grid of short tangent dashes, that rule becomes a landscape, and a solution is simply a curve that agrees with every dash it passes through. Euler's method walks the landscape the obvious way: read the slope where you are standing, go straight for one step, read again. Because a straight step always falls on the outside of a bend, the error is systematic rather than random — and it shrinks in direct proportion to the step, which is what 'first order' means. Push the step too far on the logistic equation and Euler climbs through a ceiling the equation itself forbids, which shows that a numerical method can invent behaviour the model never had. Students can change step size and starting height and compare what happens. Use it for math inquiry with High School on a projector or student device.

Standards: Common Core CCSS.MATH.CONTENT.HSF.BF.A.1.A, CCSS.MATH.CONTENT.HSF.BF.A.2

• Drag the starting point across the field and watch the same equation produce a whole family of curves
• Halve the step size and check that the gap at the end halves with it
• Set the step as large as it goes and look at where Euler's corners cut across the bends
• Switch to y(1 − y): past a step of 1 Euler climbs through the ceiling at y = 1, and past 2 it stops settling back at all
• Type your own dy/dx and find a starting point where Euler and the true curve barely part company

1. 1Predict how changing step size will affect the euler's answer, then test the prediction.
2. 2Use evidence from the live calculation and visual to explain the result.

Questions teachers ask