Curves That Obey Every Segment

You'll learnto sketch a particular solution from a given point in both directions, to find the equilibrium solutions before drawing anything at all, and to see why a curve that crosses one has stopped solving the equation.

A slope field is a set of instructions and a solution curve is what follows them. Every segment the curve passes has to be its tangent there, and that single rule settles four questions at once: where the curve starts, which way it bends, what it approaches in the long run, and the one thing it can never do. The horizontal lines along which the right side is zero are themselves solutions — constant ones — and nothing else in the picture may touch them.

8 STEPS · 6 QUICK CHECKS · AN EQUILIBRIUM IS A BARRIER, NOT A GUIDELINE · v1

dy/dx = y(2 − y) ⇒ equilibria y = 0 and y = 2 ⇒ a curve starting between them stays between them

Before you start

What you're looking at

The same lattice of short segments Topic 7.3 built, but now it is given rather than drawn. The question has changed from "what does this field look like" to "what curve is allowed to live in it".

The question

Given a field and one point, what does the solution through that point do — to the left as well as to the right, and in the long run?

Watch for

The horizontal lines drawn in teal. Each one is a constant solution, and every other curve in the picture stops short of it. That is not a drawing habit; it is the whole content of the topic.

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