A slope field draws a short segment at each point with the slope the differential equation prescribes there.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
A family of solutions
Following the segments from any starting point traces a solution curve. Different starting points give different members of the family.
The initial condition picks one
A differential equation has infinitely many solutions. An initial condition selects a single curve from the family.
Reading the field
If dy/dx depends only on x, every column looks alike. If it depends only on y, every row does.
Equilibrium solutions
Where dy/dx is zero for all x at some y value, a horizontal line is a solution and nearby curves bend toward or away from it.
Why it matters
Most differential equations cannot be solved in closed form. A slope field describes the behaviour regardless.
Seeing a solution before finding one
A slope field draws a short segment at each point with the slope the differential equation prescribes. Solution curves follow those segments, so the shape of the solutions is visible without solving anything.
Reading and drawing them
Substitute the coordinates into dy/dx to get the slope at each point. Look for where the slope is zero, where it is steep, and where it does not depend on one of the variables — those patterns identify the equation.
Sketching a particular solution
Start at the initial condition and follow the segments in both directions, staying parallel to nearby slopes. The sketch should not cross segments, and it should not cross other solution curves.
They work when the equation does not
Most differential equations have no closed-form solution. A slope field still shows the behaviour, which is why qualitative methods matter as much as analytic ones in real applications.
Step 2: Try It Yourself
Tap and try it out.
- The equationdy/dx = a·y
- Through(0, 1)
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 Ines Read a Slope Field
Ines is shown a field where every segment in a row has the same slope.
- Step 1
Segments identical along a row means the slope does not change with x.
Step 4: Your Turn
Practice makes it stick.
The Pattern
Problem 1 of 2
Every segment in a column is identical. Does dy/dx depend on x or y? 1 x only, 2 y only.
The Slope
Problem 2 of 2
dy/dx = 2y. What is the slope at the point where y = 3?
Read the Field
1 of 8
dy/dx = x. What is the slope at x = 4?
2 of 8
dy/dx = 3y. Slope where y = 2?
3 of 8
dy/dx = y. At which y is there an equilibrium solution?
4 of 8
dy/dx = y − 5. At which y is there an equilibrium?
5 of 8
How many solutions does a differential equation have without an initial condition? Enter 0 for infinitely many.
6 of 8
dy/dx = x + y. Slope at (1, 2)?
7 of 8
Sort each field description by what dy/dx depends on.
Tap something to move it.
- Empty
- Empty
8 of 8
dy/dx = 2x. Slope at x = 5?
Step 5: Quick Check
Show what you know.
Question 1 of 2
dy/dx = 4y. What is the slope where y = 3?
Question 2 of 2
What does an initial condition do?
What You Learned
- A slope field draws the prescribed slope at each point.
- Following the segments traces a solution curve.
- An initial condition selects one curve from the family.