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Cogito

Differential Equations · Chapter 3 · Lesson 3

Practice Worksheet

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

NameDate

Figure — use these to answer the problems

Notice that every horizontal row of dashes is identical. That sameness is exactly what autonomous means.
  • The equationdy/dx = −a·y
  • Through(-3, 2)
the rate1
starting x-3
starting y2
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. dy/dt = (y − 3)(y − 7) with y = 5. What is dy/dt?

    Answer

  2. What makes an equation autonomous?

    • The rate depends only on y, never explicitly on t.
    • The equation is linear.

  3. dy/dt = (y − 1)(y − 4). How many equilibria are there?

    Answer

  4. dy/dt = (y − 1)(y − 4) with y = 2. What is dy/dt?

    Answer

  5. dy/dt = (y − 1)(y − 4) with y = 6. What is dy/dt?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Is dy/dt = y + t autonomous? 1 yes, 0 no.

    Answer

  2. Is dy/dt = y² − 9 autonomous? 1 yes, 0 no.

    Answer

  3. dy/dt = y² − 9. What is the positive equilibrium?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Sort each equation by whether it is autonomous.

    Write each item under the heading it belongs to: dy/dt = 3y · dy/dt = 3t · dy/dt = y(5 − y) · dy/dt = y + sin t

    Autonomous

    Not autonomous

  2. 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

  3. The Sign

    dy/dt = (y − 2)(y − 8) with y = 5. What is dy/dt?

    Answer

  4. The Destination

    dy/dt = (y − 2)(y − 8) starting at y = 5. Which equilibrium does the solution approach?

    Answer