An equilibrium is a constant solution. Find them by setting the rate to zero and solving for y, which needs no calculus at all.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Stable equilibria
If nearby solutions move toward the equilibrium, it is stable. A small disturbance dies away.
Unstable equilibria
If nearby solutions move away, it is unstable. A pencil balanced on its point is the standard image.
Semi-stable
Some equilibria attract from one side and repel from the other. Those are semi-stable.
The phase line
Mark the equilibria on a vertical line and note the sign of the rate between them. Arrows up or down classify every one.
Why it matters
Long-run behaviour is usually the real question. A carrying capacity, a terminal velocity and a steady temperature are all stable equilibria.
Values the system sits still at
An equilibrium is a constant solution, found by setting the derivative to zero and solving. The system placed exactly there never moves, which makes them the skeleton of the long-run behaviour.
Stable and unstable
A stable equilibrium attracts nearby solutions; an unstable one repels them. Which it is can be read from the sign of the derivative either side, without solving anything.
Semi-stable equilibria exist
Some attract from one side and repel from the other. They arise where the rate function touches zero without changing sign, and they are easy to miss if only stability is checked.
They answer the question that usually matters
In applications the long-run outcome is often what is wanted, and equilibria give it directly. A full solution is more than is needed when the question is where the system ends up.
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·y
- Through(-3, 0.5)
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 Priti Classify Two Equilibria
Priti analyses dy/dt = y(4 − y), the logistic equation.
- Step 1
She sets the rate to zero, finding equilibria at y = 0 and y = 4.
Step 4: Your Turn
Practice makes it stick.
The Capacity
Problem 1 of 2
dP/dt = P(500 − P). What is the non-zero equilibrium population?
The Cooling Cup
Problem 2 of 2
dT/dt = −0.2(T − 20). What is the equilibrium temperature?
Settle or Escape
1 of 8
dy/dt = y − 7. What is the equilibrium?
2 of 8
dy/dt = y − 7. Is that equilibrium stable? 1 yes, 0 no.
3 of 8
dy/dt = 7 − y. Is the equilibrium stable? 1 yes, 0 no.
4 of 8
dy/dt = y(6 − y). How many equilibria are there?
5 of 8
dy/dt = y(6 − y). What is the stable equilibrium?
6 of 8
dT/dt = −0.5(T − 15). What is the long-run temperature?
7 of 8
Sort each behaviour by the kind of equilibrium it describes.
Tap something to move it.
- Empty
- Empty
8 of 8
A solution starting exactly at an unstable equilibrium. Does it move? 1 yes, 0 no.
Step 5: Quick Check
Show what you know.
Question 1 of 2
dy/dt = y(9 − y). What is the stable equilibrium?
Question 2 of 2
What makes an equilibrium stable?
What You Learned
- An equilibrium is a constant solution, found by setting the rate to zero.
- It is stable when nearby solutions return, and unstable when they run away.
- Carrying capacities, terminal velocities and steady temperatures are all stable equilibria.