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: Picture It
Start with something you already know.
Signposts on a hillside
- You walk across a hillside from left to right, with no map of the trail.
- Every spot you stand on has a small sign showing how steeply to climb or drop there.
- You read the sign under your feet, take a tiny step that way, and read the next one.
- Start from a different spot and the same signs lead you along a different trail.
Each sign is the slope the differential equation gives at that point, and a slope field draws all of them.
The trail you walk is a solution curve, and the spot where you start picks which one.
Where the picture breaks: real signs stand a few steps apart, but the equation gives a slope at every single point, so a solution curve bends smoothly instead of in corners.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
You have no formula for a function y. You only know that dy/dx = x − y², and that its graph passes through the point (5, 2). How steep is the graph at that point?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
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 3: 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 4: 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 5: Watch an Example
One step at a time.
Watch Noor Draw a Field and Follow It
A fresh equation, dy/dx = 3 − y. Noor makes a short table of slopes, finds the flat line, and follows two solutions, without solving anything.
- Step 1
The equation dy/dx = 3 − y has no x in it. Moving sideways changes x but not y, so it cannot change the slope: every point in a row has the same dash, and one slope per row is the whole table.
Watch Leo Fill a Field with Isoclines
Leo draws the field of dy/dx = y − x using lines of equal slope instead of a long table. He sets up the idea; one slope, one isocline, and what one special line turns out to be are yours.
- Step 1
Leo’s equation, dy/dx = y − x, uses both letters, so the slope changes both across and up the page. The slope is some number c wherever y − x = c, which is the straight line y = x + c. Every isocline of this field is a line parallel to y = x.
Step 6: Your Turn
Practice makes it stick.
The Growing Savings
Problem 1 of 17
A savings account earns 4% interest a year, and its owner also deposits $200 a year, spread evenly. The balance B, in dollars, follows dB/dt = 0.04B + 200, with t in years. At the moment the balance is $5,000, how fast is it growing, in dollars per year?
dollars per year
The Steady Drip
Problem 2 of 17
A drip gives a patient 30 milligrams of a medicine each hour, while the body clears 10% of the amount present each hour. The amount M, in milligrams, follows dM/dt = 30 − 0.1M. At what amount does the level stay steady, neither rising nor falling?
milligrams
Follow the Dashes
Each one is a slope at a point, a flat line, an isocline, or where a curve settles. The equation gives every slope; nothing needs solving.
Problem 3 of 17
In the slope field of dy/dx = x² + y, what is the slope of the dash at (2, −1)?
Problem 4 of 17
The field of dy/dx = 8 − y³ has one row of flat dashes. At what height y is that equilibrium?
Problem 5 of 17
For dy/dx = 2x − y, every dash has slope 4 along one straight line. Write that isocline in the form y = …
Problem 6 of 17
How many equilibrium solutions does the field of dy/dx = y² − 9 have?
Problem 7 of 17
In the field of dy/dx = 6 − 3y, the solution through (0, 0) rises and settles toward a horizontal line. At what height y?
Whose Step?
Each page has a slip. Find the first line that does not hold, then put it right.
Problem 8 of 17
Priya found the slope of the field of dy/dx = y − x² at the point (3, 2). Which line is the first that does not hold?
Priya’s slope of dy/dx = y − x² at (3, 2)
- 1x = 3 and y = 2
- 2y − x² = 2 − 6
- Answerslope = −4
Problem 9 of 17
What is the slope of the field of dy/dx = y − x² at (3, 2)?
Problem 10 of 17
Sam found the isocline of slope 2 for dy/dx = x + y. Which line is the first that does not hold?
Sam’s isocline of slope 2 for dy/dx = x + y
- 1slope 2 where x + y = 2
- 2y = 2 + x− x
- Answerisocline: y = x + 2
Problem 11 of 17
For dy/dx = x + y, write the isocline of slope 2 in the form y = …
Problem 12 of 17
Ana looked for an equilibrium in the field of dy/dx = x − 3. Which line is the first that does not hold?
Ana’s search for an equilibrium of dy/dx = x − 3
- 1slope 0 where x − 3 = 0
- 2that is, where x = 3
- 3every dash on the vertical line x = 3 is flat
- Answerso y = 3 is an equilibrium solution
Problem 13 of 17
How many equilibrium solutions does dy/dx = x − 3 have?
Pick the Move
Decide what to reach for first. Neither needs the equation solved.
Problem 14 of 17
You want a quick hand sketch of the whole field of dy/dx = x + 2y. Which move fills it fastest?
Problem 15 of 17
Ravi wants to know where the solution of dy/dx = y² − 4y that starts at (0, 1) goes in the long run. Which move should come first?
Same Field?
Judge these from the equations alone. No field needs drawing.
Problem 16 of 17
Select every pair of equations that draw exactly the same slope field.
Problem 17 of 17
Select every equation whose dashes are flat all along the line y = x.
Step 7: 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?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
A slope field comes from dy/dx = ax + by, for two numbers a and b. The dash at (1, 0) has slope 3, and the dash at (0, 1) has slope −2. What is the slope of the dash at (2, 2)?
Stretch 2 of 2
In the field of dy/dx = x + y, the isocline of slope c is the line y = c − x. For exactly one value of c, that isocline is itself a solution curve. What is c?
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.
For the grown-up
The misconception this lesson targets is treating any line of flat dashes as an equilibrium. An equilibrium is a horizontal line y = k on which the slope is 0 for every x; a field like dy/dx = x − 3 has flat dashes all the way up the vertical line x = 3 and no equilibrium at all, because a solution is a function of x and cannot run straight up the page. It comes from setting the right-hand side to zero and solving for whichever letter appears. A student who asks "is this line horizontal, and is it flat for every x?" will not make it. Two close cousins: taking an isocline for a solution curve (it is one only when the line’s own slope equals the slope it carries, as with y = x + 1 for dy/dx = y − x), and reading a dash as the height of the solution rather than its steepness.
A note on the theory’s second curve: the point (−2, 1) lies on y = −1 − x, which is itself a solution (C = 0 in y = Ceˣ − x − 1, the line found in Ready for more), so the curve through it falls in a straight line forever rather than leveling off. The curve through (−2, 2), where the slope is 0, is flat there and then rises: it is y = e^(x + 2) − x − 1.