Lesson · Differential Equations and Euler's Method · Pictures of solutions

Slope Fields

An equation y′ = f(t, y) gives a slope at every point of the plane, and that is all the information it has. Draw a short segment of that slope at each point of a grid and the equation is a picture, with no solving involved. The slope at a chosen point is a number to compute exactly, and the points where it is zero make a curve to find.

Open the interactive lesson Practice

y′ = f(t, y) gives a slope at each point slope at (t, y) is f(t, y), computed exactly y′ = t − y at (2, 0): slope 2 slope 0 on y = t, the nullcline a segment shows slope, not where it goes

A slope field: the equation as a grid of slopes

Core ideas

What this lesson establishes

Three ideas: what the picture is made of, how one slope is computed, and the one curve worth finding straight away.

The equation is a rule that assigns a slope to every point

In y′ = f(t, y) the right side takes a point (t, y) and returns a number. A solution is a curve whose slope at each of its points is that number, and so the equation is a statement about slopes before it is a statement about functions. A slope field draws that statement: a short segment with slope f(t, y) at each point of a grid.

A slope is computed by substitution, and it is exact

To find the slope at (2, 0) for y′ = t − y, put t = 2 and y = 0: 2 − 0 = 2. There is no limit and no approximation, because the slope is the value of a polynomial at a point. The lab prints it as a fraction when it is one.

The nullcline is where the slope is zero

Set f(t, y) = 0 and solve. For y′ = t − y that is y = t, a line. Along it every segment is flat, and a solution that reaches it has a horizontal tangent there. Segments above the nullcline slope one way and segments below slope the other, which is why it organises the whole picture.

The material

Drawing an equation without solving it

The grid, the slope at one point, the zero-slope curve, and a field in which the formula tells you a shortcut.

Definition · Slope field and nullcline

For the equation y′ = f(t, y), the slope field is the collection of short segments, one centred at each point of a grid, whose slope is f(t, y) at that point.

The nullcline is the set of points where f(t, y) = 0, the points whose segment is horizontal.

Take y′ = t − y. At (2, 0) the slope is 2 − 0 = 2, so the segment rises two for every one across. At (1, 1) it is 1 − 1 = 0, so the segment is flat. At (0, 2) it is 0 − 2 = −2, and the segment falls steeply. Three substitutions, three segments, and a grid of fifteen by fifteen of them is that arithmetic repeated.

y′ = t − y point (t, y) slope t − y (2, 0) 2 (1, 1) 0 (0, 2) −2 (3, 1) 2

The nullcline of y′ = t − y is found by setting the right side to zero: t − y = 0, so y = t. Every point on that line, such as (1, 1) and (−2, −2), has a flat segment. The lab prints it in the tile named Slope 0.

Fields that depend on one variable

Two special cases are worth recognising because the picture is then regular. For y′ = y, the slope depends on y alone: at (0, 1) it is 1, and it is 1 at every other point at height 1, so the segments in each row are parallel copies. The nullcline is the line y = 0.

For y′ = 2t the slope depends on t alone: at (1, 2) it is 2, and it is 2 on the whole vertical line t = 1. The nullcline is the line t = 0, and the lab prints it as none in y because no height y makes the slope zero for every t; there is no equation to solve for y. This is the equation of “What a Differential Equation Is”, and its solutions t² + C are vertical shifts of one another, which is exactly what a field of identical columns would let you see.

Example · Reading one segment correctly

At (2, 0) on y′ = t − y the segment has slope 2.

That says a solution through (2, 0) is rising there at a rate of 2. It does not say the solution goes on to (3, 2): by then the point is a different point with a different slope, 3 − 2 = 1, and the curve has bent. A segment is a tangent, the direction of travel at that instant.

About the drawing: the segments and any curve are plotted with floating-point numbers, and the legend says a solution is drawn. The slope tile is computed exactly. If a number matters, read it from the tile.

Interactive

Slope fields

The slope at every point, before any solving

A slope field, one exact slope, and the nullcline

The segments are drawn at the grid points; the slope tile is the exact value of f at the point you type. Pick each preset, find the slope by hand first, and then compare. Change the point in the first preset to (2, 0) and expect a slope of 2.

Equation

f(t, y) in y' = f(t, y)

Window: tmin tmax ymin ymax

Start of a drawn solution: t0 y0 (may be empty)

Point: t y

Segments per side

Slope at the point—

Direction there—

Slope 0 (nullcline)—

Horizontal solutions—

Type a polynomial in t and y: t - y, y(1 - y), y^2, 2t.

Method

Building the picture of an equation

From a right-hand side to a field you can read.

Choose a window and a grid

Decide the range of t and of y to show, and a grid of points inside it. A finer grid shows more detail and the same information.

Compute the slope at a point by substituting

Put the t and y of the point into f. Do it exactly, as a fraction if need be, and write the number beside the point.

Draw the segment through the point

A segment with that slope: rise m for a run of 1. Flat for 0, rising for a positive number, falling for a negative one.

Find where the slope is zero

Set f(t, y) = 0 and solve for y where you can. That curve separates rising segments from falling ones, and it is the first thing to mark.

Slopes, segments and the nullcline

Score 0 / 0

Choose the answer you can justify.

y′ = t − y at three points, and its nullcline

Compute the slopes at (2, 0), (1, 1) and (0, 2), then find where the slope is zero.

f(t, y) = t − y (2, 0): 2 − 0 = 2 rising (1, 1): 1 − 1 = 0 flat (0, 2): 0 − 2 = −2 falling slope 0: t − y = 0, so y = t above the line, y > t: t − y is negative, falling below the line, y < t: t − y is positive, rising

The three points straddle the line y = t: (2, 0) is below it with a positive slope, (0, 2) is above it with a negative slope, and (1, 1) is on it with slope zero. Every solution crosses that line horizontally, which is a statement about the whole family and was read from a single equation, t − y = 0.

For a rehearsal, work out the slope at (3, 1) yourself, then look at the preset: 3 − 1 = 2.

Common mistakes

Reading a segment as where the solution goes next, rather than its slope thereThe segment at (2, 0) on y′ = t − y has slope 2 and it is a tangent. The slope at the next point of the curve is a different number from the field, so the curve bends, and joining segments end to end is Euler's method, which the stepping half of this course shows is only approximate.

Confusing the slope with the value of yThe number the field gives at (1, 1) is 0, the slope, and the point is at height 1. A flat segment at a high point and a steep segment at a low point are both normal.

Expecting the nullcline to be a solutionWhere f = 0 the segments are flat, but a solution that touches the line y = t carries on and leaves it, because the slope changes with the point. A horizontal solution needs f = 0 along the whole line y = c for every t; that is a stronger condition.

Check your understanding

✓

Finish when you can compute a slope at a point exactly, draw its segment, and find the nullcline.

You should be able to substitute a point into f(t, y) and write the slope as a fraction or integer, sketch the segment for a positive, negative or zero slope, solve f(t, y) = 0 for the nullcline of a right side that is linear in y, and say what a segment shows and what it does not.

Note: The picture is drawn. The next lesson reads it: which curves are solutions that stay put, where solutions head, and what can never happen, and also what the picture can only suggest. “Reading a Slope Field” does it.