A first-order equation gives dy/dx at every point. Draw a short dash of that slope at each point and the plane fills with directions: a slope field. Any solution curve must be tangent to the dashes it passes through, so starting anywhere, the field says where to go next. It works even when no formula exists, and equilibria and isoclines make it quick to draw.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Drawing one
dy/dx = x + y: at (0, 0) the slope is 0; at (1, 0), 1; at (0, 1), 1; at (2, 0), 2; at (−1, 1), 0; at (1, −1), 0. A small table of points and slopes, then a dash at each. Six dashes already show the pattern.
Isoclines
Where is the slope 0? Wherever x + y = 0: the line y = −x. Slope 1: the line y = 1 − x. Each isocline is a line of equal slope, and drawing three or four of them fills the field faster than any table.
Following a solution
From (0, 0) the slope is 0, so the curve starts flat. Moving right, x grows, so the slope grows and the curve bends upward, faster and faster. From (−2, 1) the slope is −1: it falls first, then levels off near y = −x, then rises.
A check
The solution through (0, 0) is y = eˣ − x − 1: at 0 it gives 0, and y′ = eˣ − 1 = (eˣ − x − 1) + x = y + x. Flat at the start, then rising ever faster, exactly as the field predicted.
Equilibria
Where the dashes are horizontal for every x, the derivative is zero along a whole line: a constant solution. dy/dx = −y has one at y = 0. dy/dx = x + y has none, because the slope depends on x.
No formula needed
dy/dx = sin(xy) has no closed-form solution, but its field can be drawn and its solutions traced. Most real equations are like that, and the picture is how they are understood.
Step 2: Try It Yourself
Tap and try it out.
For dy/dx = x + y, tabulate slopes at six points, find the isocline of slope 0, look for equilibria, and describe the solution through (0, 0)
- 1(0, 0): 0; (1, 0): 1; (0, 1): 1; (2, 0): 2; (−1, 1): 0; (1, −1): 0slope = x + y at each point
Step 0 of 4
- The equationdy/dx = −a·y
- Through(-3, 3)
Every short line shows the slope the equation demands at that point. The curve simply follows them, and the marked point is the initial condition that picks it out of the family.
- The equationdy/dx = a·(x + y)
- Through(0, 0)
Every short line shows the slope the equation demands at that point. The curve simply follows them, and the marked point is the initial condition that picks it out of the family.
Step 3: In Real Life
A weather map
A slope field draws the direction a solution moves at every point, like the arrows on a wind map. Meteorologists trace forecast tracks through them before solving anything.
Step 4: Watch an Example
One step at a time.
Watch Elif Read a Field
Elif studies dy/dx = −y without solving it.
- Step 1
Above the axis y is positive, so the slope is negative and solutions fall. Below it y is negative, so the slope is positive and solutions rise.
Step 5: Your Turn
Practice makes it stick.
The Slope
Problem 1 of 2
dy/dx = x + y. What is the slope of the field at the point (2, 3)?
The Flat Line
Problem 2 of 2
dy/dx = 2y. At what value of y is the slope zero?
Follow the Dashes
1 of 8
dy/dx = x. What is the slope at (4, 9)?
2 of 8
dy/dx = y. What is the slope at (4, 9)?
3 of 8
dy/dx = xy. What is the slope at (3, 2)?
4 of 8
dy/dx = −y. What is the slope at (0, 5)?
5 of 8
dy/dx = 3. Are all the dashes parallel?
6 of 8
dy/dx = y − 4. At what value of y is the equilibrium?
7 of 8
Match each equation to what its field looks like.
Tap a card on the left to start.
8 of 8
A solution starting exactly at an equilibrium. How far does it move?
Step 6: Quick Check
Show what you know.
Question 1 of 2
dy/dx = x + y. What is the slope at (1, 6)?
Question 2 of 2
Why is a slope field useful?
What You Learned
- A slope field draws the required slope at every point of the plane.
- Solution curves must run tangent to the dashes they pass through.
- Isoclines are lines of equal slope, and equilibria are horizontal lines of zero slope.
- Growth, decay and equilibria are all visible without solving anything.