AP® Calculus AB · Unit 7 · Topic 7.4
Matching a Slope Field to Its Differential Equation
Matching a slope field to its equation means testing a few specific points: plug each point's x and y into the candidate right-hand side and check if the result matches the drawn slope.
A slope field is a grid of little dashes, each dash showing the slope dy/dx that the differential equation predicts at that exact point. To match a field to its equation, you don't need to study the whole picture — you need a few well-chosen points where the behavior is unmistakable. The cleanest clues come from axes. If every dash along the y-axis is flat (slope 0), that tells you the right-hand side must equal 0 whenever x = 0, no matter what y is. Only an equation built purely from x, like dy/dx = x, guarantees that: plug in x = 0 and you get 0 automatically, for every y.
The general method: pick a labeled point (x0, y0) in the field, read off the slope drawn there, then plug x0 and y0 into each candidate equation's right-hand side. Whichever candidate spits out the matching number survives. Do this at two or three points, not just one, because steepness patterns alone can fool you — dy/dx = x and dy/dx = y can both look 'steep away from one axis and flat near another' at a glance, but they disagree at specific points like (1, 2) versus (2, 1). You need actual numbers, not just a visual impression, to be sure.
This connects directly to separation of variables: once you've identified the correct equation from its field, you can solve it algebraically and check that solution curves follow the dashes. Swapping x and y in an equation also swaps the roles of the two axes in the picture — for dy/dx = x - y versus dy/dx = y - x, every slope flips sign, so a field that tilted one way now tilts the mirror-opposite way at the same point.
Key terms
- slope field
- A grid of short line segments showing the slope dy/dx that a differential equation predicts at each plotted point.
- differential equation
- An equation relating a function to its derivative, here written as dy/dx equals some expression in x and y.
- right-hand side
- The expression in x and y that dy/dx is set equal to, which determines the slope at every point.
- separation of variables
- A solving technique that works once the correct differential equation has been identified from its slope field.
Watch it explained
- How it worksHow do you test candidate equations against a few key points in the field?
- Worked exampleFind which of dy/dx = x, dy/dx = y, or dy/dx = x + y makes every slope flat along the y-axis.
- Common mistakeWhy isn't the steepness pattern alone enough to identify the equation?
- Going furtherWhat changes in the picture when you swap x and y in the differential equation?
Check yourself
End of topic.