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Cogito

Differential Equations · Chapter 1 · Lesson 3

Practice Worksheet

Equilibria and Stability

The values a system settles into, and the ones it runs from.

12 problems · about 23 minutes · F-IF.B.4, F-IF.C.7

NameDate

Figures — use these to answer the problems

Start a solution above the axis and then below it. Both approach y = 0, which is what stable means.
  • The equationdy/dx = −a·y
  • Through(-3, -3)
the rate1
starting x-3
starting y-3
Figure 1
Now dy/dx = y. Start just off the axis either way and the solution runs away from it. That is instability.
  • The equationdy/dx = a·y
  • Through(-3, 0.5)
the rate1
starting x-3
starting y0.5
Figure 2

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. dy/dt = y(9 − y). What is the stable equilibrium?

    Answer

  2. What makes an equilibrium stable?

    • Nearby solutions move back toward it.
    • It happens to be at zero.

  3. dy/dt = y − 7. What is the equilibrium?

    Answer

  4. dy/dt = y − 7. Is that equilibrium stable? 1 yes, 0 no.

    Answer

  5. dy/dt = 7 − y. Is the equilibrium stable? 1 yes, 0 no.

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. dy/dt = y(6 − y). How many equilibria are there?

    Answer

  2. dy/dt = y(6 − y). What is the stable equilibrium?

    Answer

  3. dT/dt = −0.5(T − 15). What is the long-run temperature?

    Answer

Stretch Yourself

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

4 problems

  1. Sort each behaviour by the kind of equilibrium it describes.

    Write each item under the heading it belongs to: Nearby solutions return to it · Nearby solutions run away from it · A small disturbance dies away · A small disturbance grows

    Stable

    Unstable

  2. A solution starting exactly at an unstable equilibrium. Does it move? 1 yes, 0 no.

    Answer

  3. The Capacity

    dP/dt = P(500 − P). What is the non-zero equilibrium population?

    Answer

  4. The Cooling Cup

    dT/dt = −0.2(T − 20). What is the equilibrium temperature?

    Answer