Cogito
Differential Equations · Chapter 3 · Lesson 3
Autonomous Equations and the Phase Line
When the rate depends only on the state.
12 problems · about 23 minutes · F-IF.B.4, F-IF.C.7
What this lesson teaches
The student analyses autonomous equations with a phase line and predicts long-run behaviour.
- An autonomous equation has dy/dt depending only on y, never on t directly.
- The whole picture compresses to a phase line marked with equilibria and arrows.
- Long-run behaviour follows from the signs alone, with no integration at all.
Warm Up
Straightforward practice. Get the method working first.
5 problemsdy/dt = (y − 3)(y − 7) with y = 5. What is dy/dt?
Answer -4
Why −4.
What makes an equation autonomous?
Answer The rate depends only on y, never explicitly on t.
Why No explicit time dependence.
dy/dt = (y − 1)(y − 4). How many equilibria are there?
Answer 2
Why Two factors.
dy/dt = (y − 1)(y − 4) with y = 2. What is dy/dt?
Answer -2
Why 1 times −2.
dy/dt = (y − 1)(y − 4) with y = 6. What is dy/dt?
Answer 10
Why 5 times 2.
Build It Up
The same ideas with more to keep track of.
3 problemsIs dy/dt = y + t autonomous? 1 yes, 0 no.
Answer 0
Why Time appears explicitly.
Is dy/dt = y² − 9 autonomous? 1 yes, 0 no.
Answer 1
Why Only y appears.
dy/dt = y² − 9. What is the positive equilibrium?
Answer 3
Why Set y² = 9.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each equation by whether it is autonomous.
Answer Autonomous: dy/dt = 3y, dy/dt = y(5 − y) · Not autonomous: dy/dt = 3t, dy/dt = y + sin t
Why Look for t on the right-hand side.
A solution starting exactly at an equilibrium of an autonomous equation. Where does it end up? Enter the equilibrium value if it stays, using 0 for an equilibrium at zero.
Answer 0
Why It never moves.
The Sign: dy/dt = (y − 2)(y − 8) with y = 5. What is dy/dt?
Answer -9
Why −9.
The Destination: dy/dt = (y − 2)(y − 8) starting at y = 5. Which equilibrium does the solution approach?
Answer 2
Why 2.