A first-order equation gives dy/dx at every point. Drawing a short dash of that slope at each point fills the plane with directions.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Solutions follow the dashes
Any solution curve must be tangent to the dashes it passes through. Starting anywhere, the field tells you where to go next.
No formula needed
The picture exists even when no formula can be found. Most real equations have no closed-form solution, and this still works.
What you can read off
Whether solutions grow, decay, level off or blow up is visible immediately from the field.
Equilibria
Where the dashes are horizontal, the derivative is zero. A solution starting there never moves.
Isoclines
An isocline joins the points with one common slope. Sketching a few makes a field quick to draw by hand.
The shape of every solution at once
A slope field draws a short segment at each point with the slope the equation prescribes there. Solution curves follow those segments, so the family of solutions becomes visible before any is found.
Reading a field
Look for where the slope is zero, where it is steep, and whether it depends on only one of the variables. Those features identify the equation and predict the long-run behaviour of solutions.
Solution curves never cross
Where the uniqueness theorem applies, two solution curves cannot intersect — a crossing point would have two different solutions through it. That constraint is a useful check when sketching.
They work when solving fails
Most differential equations have no closed-form solution. A slope field still shows what solutions do, which is why qualitative methods sit alongside analytic ones rather than beneath them.
Step 2: Try It Yourself
Tap and try it out.
- 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: Watch an Example
One step at a time.
Watch Elif Read a Field
Elif studies dy/dx = −y without solving it.
- Step 1
She checks the sign above the axis, where y is positive, so the slope is negative and solutions fall.
Step 4: 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? 1 yes, 0 no.
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 5: 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.
- Growth, decay and equilibria are all visible without solving anything.