Definition: A slope field is a grid of short line segments showing the slope that a differential equation assigns at each point, revealing the shape of its solutions.
What is a slope field?
For dy/dx = f(x, y), plug the coordinates of each grid point into the formula and draw a tiny segment with that slope. Solution curves follow the segments like a boat following a current.
How do you read one?
Worked example: dy/dx = x – y.
- At (2, 1): slope = 2 – 1 = 1
- At (0, 3): slope = 0 – 3 = -3
- At (1, 1): slope = 0, so the segment is flat
Every point on the line y = x has slope 0. Above that line the slopes are negative, and below it they are positive.
Sketch a solution from the starting point (0, 3): it begins falling steeply, then flattens and approaches the line y = x – 1.
What mistake occurs?
Letting a sketched curve cross a segment at the wrong angle. A solution must be tangent to every segment it touches. Mathbot also reminds you that the slope depends on both x and y, so segments in one column can differ.