# Math Insight

### Quiz 10

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Total points: 8
1. Consider the differential equation $\diff{ w }{t}=h(w)$ where the function $h(w)$ is shown below. All the zeros of $h$ (i.e,. the values of $w$ where $h(w)=0$) are integers.

Find all the equilibria and determine their stability.

The equilibria are
. (If there are more than one equilibrium, enter them in increasing order, separated by commas.)

Stability of equilibria:
(Specify the stability of each equilibrium in the same order as above. Enter stable if an equilibrium is stable or unstable if it is unstable. Separate multiple answers by commas.)

For example, if there are four equilibria and they are, in order, stable, unstable, unstable, and unstable, then you should enter stable, unstable, unstable, unstable in the answer blank.

2. Consider the differential equation \begin{align*} \diff{ z }{t} & = - 3 z^{2} + 30 z - 27 \\ z_0 & = 7\\ \end{align*}

Find all the equilibria and determine their stability.

The equilibria are
. (If there are more than one equilibrium, enter them in increasing order, separated by commas. If there are no equilibria, enter none.)

Stability of equilibria:

Specify the stability of each equilibrium in the same order as above. Enter stable if an equilibrium is stable or unstable if it is unstable. Separate answers by commas. If there are no equilibria, enter none.

For example, if there are four equilibria and they are, in order, stable, unstable, unstable, and unstable, then you should enter stable, unstable, unstable, unstable in the answer blank.

3. Consider the differential equation \begin{align*} \diff{ v }{t} & = - 3 v \left(v^{2} - v + 3\right) \\ v_0 & = 7\\ \end{align*}

Find all the equilibria and determine their stability.

The equilibria are
. (If there are more than one equilibrium, enter them in increasing order, separated by commas. If there are no equilibria, enter none.)

Stability of equilibria:

Specify the stability of each equilibrium in the same order as above. Enter stable if an equilibrium is stable or unstable if it is unstable. Separate answers by commas. If there are no equilibria, enter none.

For example, if there are four equilibria and they are, in order, stable, unstable, unstable, and unstable, then you should enter stable, unstable, unstable, unstable in the answer blank.

4. Consider the dynamical system $u'(t)=m(u)$ where the function $m(u)$ is shown below. All the zeros of $m$ (i.e,. the values of $u$ where $m(u)=0$) are integers.

Use the below applet to sketch the phase line, including equilibria and direction field. Use a solid circle for stable equilibria and an open circle for unstable equilibria.

Feedback from applet
equilibria:
number of equilibria:
stability of equilibria:
vector field:

Drag the slider labeled $n_e$ to specify the number of equilibria. Drag the red points to the location of the equilibria. You can click on a point to change it between open and solid. Click the segments between equilibria to show a direction field. Clicking the segment again changes the direction of the vectors.