Direction Fields

Main idea

The equation $y' = f(t, y)$ tells you the slope a solution must have at every point, so you can draw those slopes as little segments and watch the solutions take shape. That lets you see where solutions end up before you've solved anything.

y' = {\htmlData{sym=0}{f(t, y)}}

What the symbols mean

$f(t, y)$
Slope
The slope the solution through (t, y) must have there.

Explanation

A direction field is simply a picture of the slopes an equation demands: at each point of a grid, you draw a tiny segment tilted at the slope a solution passing through that point would have. It's also called a slope field; "direction" because each segment shows which way a solution is heading as it passes through. It lets you see how solutions behave, and where they end up, without solving anything. All we're really saying with $y' = f(t, y)$ is that $y'$ (the slope of a solution) equals $f(t, y)$ (a rule that turns any point into a number). Feed in a point, get a slope, draw it; that's all there is to it, so bear with it.

Remember the population example from the first lesson: $P' = 0.5P$ didn't tell us $P$ , but it told us the slope at every point. That's true of any equation $y' = f(t, y)$ . Solution curves follow these slopes like iron filings lining up in a magnetic field.

Sketching: pick grid points, compute $f(t, y)$ , and draw a tiny segment with that slope. Then trace curves that stay tangent to the segments.

Isoclines are curves where the slope is constant: solve $f(t, y) = c$ . The isocline $f = 0$ shows where solutions are flat.

A common mistake. It's tempting to think the isocline $f = 0$ is itself a solution, because every segment along it is flat. Usually it isn't. For $y' = y - t$ the segments are flat along the line $y = t$ , but that line has slope $1$ , so a solution can't run along it; solutions just cross it while they're momentarily level. A curve of flat segments is only a solution when it's flat too, which means a horizontal line.

Equilibrium solutions. Those horizontal lines are exactly what you get when $f$ depends only on $y$ (an autonomous equation). The zeros of $f$ give constant solutions. Nearby solutions either approach them (stable) or move away (unstable).

Direction fields predict where solutions end up, which is often the question that matters most.

See it

This is the slope field of y′ = (2 − y)(y + 1)². As you can see, the segments tilt up everywhere below y = 2 and down above it, so solutions crowd toward y = 2 from both sides. At y = −1 the segments go flat, but they tilt up on both sides of it: solutions below rise into that line, and solutions above leave it.

These are actual solutions of y′ = y(1 − y). Whether a curve starts barely above 0 or well above 1, it ends up at y = 1, the stable equilibrium. Notice that none of the curves starting above 0 ever settles back down to y = 0.

Worked examples

Example 1

Describe the solutions of $y' = 1 - y$ .

1
The slope $1 - y$ is positive below $y = 1$ , negative above it, and zero on it.

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

For $y' = t^2 - y$ , what is the slope at $(1, 2)$ ?

Problem 2+5 XP

Find the equilibrium solutions of $y' = y(3 - y)$ .

Problem 3+5 XP

Which equilibrium of $y' = y(3 - y)$ is stable?

Problem 4+5 XP

The solution of $y' = 2 - y$ with $y(0) = 10$ approaches what value?

Problem 5+5 XP

Where are the slopes of $y' = y - t$ zero?

Problem 6+5 XP

A skydiver's velocity obeys $v' = 9.8 - 0.2v$ (in m/s). Without solving the equation, what speed does she approach?

Problem 7+5 XP

The first picture in this lesson is the slope field of $y' = (2 - y)(y + 1)^2$ . For which values of $y$ are solutions increasing?

Problem 8+5 XP

For the same equation, give your best guess for $\lim\limits_{t\to\infty} y(t)$ when $y(0) = -1.001$ , and when $y(0) = -0.999$ , in that order.

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

For $y' = y(y - 2)(y - 4)^2$ , find $\lim\limits_{t\to\infty} y(t)$ for the starting values $y(0) = -1$ , $y(0) = 1$ and $y(0) = 3$ , in that order.

Where it's used

Ecologists read fisheries' long-term futures from direction fields of harvesting models, finding stable populations and tipping points without solving anything.

Logistic Population GrowthDiff Eq · Biology A Skydiver's Terminal VelocityDiff Eq · Physics

Integrals as Solutions Linear Equations