Slope Fields

You’ll be able to

Interpret a slope field as the geometry of a differential equation
Sketch short segments from dy/dx at sample points
Match a differential equation to its slope field and trace solution curves

A picture of a differential equation

A differential equation like dy/dx = x + y gives the slope of a solution at every point, without telling you the solution outright. A slope field visualizes this: at a grid of points you draw a short segment whose slope is the value of dy/dx there. The field shows the "flow" that every solution curve must follow — solutions thread through the segments tangentially.

Reading and building the field

To build a slope field, plug each point’s coordinates into dy/dx and draw a tiny segment with that slope. Where dy/dx = 0 the segments are horizontal; where dy/dx is large the segments are steep. Patterns reveal the equation: if the slope depends only on x, every column of segments is identical; if only on y, every row is identical. Matching those patterns is a common multiple-choice task.

Solution curves follow the flow

A particular solution through a given initial point is the curve that stays tangent to the segments as it moves. Starting at the initial condition, you follow the slopes — going right, the curve rises where segments tilt up and falls where they tilt down. The slope field lets you sketch the qualitative behavior of a solution even when you cannot solve the equation algebraically.

Slope at a point

At (x₀, y₀), the segment slope is dy/dx evaluated at (x₀, y₀)

Horizontal segments where dy/dx = 0; steeper segments where |dy/dx| is large. Slope depends only on x ⇒ identical columns.

Worked example

For dy/dx = x − y, find the slope of the field at the points (0, 0), (2, 1), and (1, 3).

1.The slope at any point is just x − y.
2.At (0, 0): slope = 0 − 0 = 0, a horizontal segment.
3.At (2, 1): slope = 2 − 1 = 1, a segment rising at 45°.
4.At (1, 3): slope = 1 − 3 = −2, a steep downward segment.

Answer: Slopes are 0, 1, and −2 respectively. Along the line y = x the slope is always 0, so the field has a row of horizontal segments there.

Tip

To match an equation to a field fast, test a few easy points. Look for where the slope is zero: dy/dx = 0 marks a whole curve of horizontal segments, and that curve’s shape (a line, a parabola) often identifies the equation immediately.

Checkpoint

For the differential equation dy/dx = 2x, what does the slope field look like?

Checkpoint

For dy/dx = y, where are the slope-field segments horizontal?

On the exam

When asked to sketch a solution curve on a given slope field, start at the initial point and keep your curve tangent to the nearby segments, never crossing them at odd angles. Follow the flow smoothly in both directions from the starting point.

Answer the 2 checkpoints as you read.