Unit 7: Differential Equations

Slope Fields

Sketching and interpreting a slope field for a differential equation.

Advanced25 min lesson4 min readUpdated August 12, 2026Author not yet attributed

Visualizing an Equation Written in Terms of a Slope

dy/dx = x − y describes a slope at every point in the plane, without giving y as a function of x directly. Before reading on: if you computed that slope at a grid of points and drew a short line segment with that exact slope at each one, would a pattern — maybe even a recognizable curve — start to emerge?

Definition — Slope Field

A slope field for a differential equation dy/dx = f(x, y) draws a short line segment of slope f(x, y) at each point (x, y) of a grid. A solution curve to the equation follows the local slope direction at every point it passes through — sketching a curve that stays tangent to the field everywhere is how a solution is found graphically, without solving the equation algebraically.

Worked Example — Building a Slope Field from a Table

For dy/dx = x − y, compute the slope at each point of a 3×3 grid: (0,0): 0. (1,0): 1. (2,0): 2. (0,1): −1. (1,1): 0. (2,1): 1. (0,2): −2. (1,2): −1. (2,2): 0. Notice every point on the diagonal y = x gives slope 0 — (0,0), (1,1), and (2,2) are all flat. That's a strong hint: check whether y = x − 1 (a line parallel to that diagonal) might be an exact solution. Its derivative is dy/dx = 1, and x − y = x − (x − 1) = 1 — they match for every x, confirming y = x − 1 genuinely solves the equation, found by reading the slope field's pattern rather than solving anything algebraically.

	y = 0	y = 1	y = 2
x = 0	0	−1	−2
x = 1	1	0	−1
x = 2	2	1	0

Slope dy/dx = x − y at each grid point — flat (0) exactly along the diagonal y = x.

Worked Example — Matching a Slope Field's Pattern to Its Equation

Contrast dy/dx = x − y with dy/dx = y. For dy/dx = y, the slope is 0 wherever y = 0 — the entire x-axis is flat, meaning y = 0 itself is a solution (an equilibrium: if you start there, dy/dx = 0 keeps you there). Above the x-axis, slopes are positive and grow steeper as y increases; below, slopes are negative and steepen downward. This produces a visually distinct field from x − y's diagonal-flat pattern — a slope field's overall shape (which lines or curves are flat, how steepness varies) is often enough to match it to its generating equation without any algebra.

Tip

Look first for where the slope field is flat (slope 0) — those points satisfy f(x, y) = 0, and if they trace out a recognizable curve, that curve is very likely an exact solution worth checking algebraically.

Common Mistakes

Assuming a slope field's segments show the actual path of a solution curve, rather than just the local direction at each point.
Each segment only shows the instantaneous slope at that single point — a solution curve has to be sketched by following the changing direction continuously from point to point, not by connecting the segments themselves.
Computing dy/dx = x − y using only the x-coordinate, ignoring the y-value at that grid point.
The slope at a point genuinely depends on both coordinates — (1, 0) and (1, 2) have very different slopes (1 and −1) for the same equation, since y enters the formula too.

Key Takeaways

A slope field draws the local slope dy/dx = f(x, y) at each point of a grid, visualizing a differential equation without solving it.
A solution curve stays tangent to the slope field at every point along its path.
Points where the slope field is flat satisfy f(x, y) = 0 and can reveal an exact solution by inspection, worth verifying algebraically.

Summary

Slope fields visualize a differential equation's solutions. The next lesson solves many differential equations exactly, using integration.

Track your progress

Back to Unit 7: Differential Equations Separable Differential Equations→