5.17 Phase portraits Quiz

1. A system has eigenvalues $-2$ and $3$. What is the origin?

• A. A stable node
• B. An unstable node
• C. A saddle point
• D. A centre

2. A system has eigenvalues $1\pm2i$. Describe the trajectories.

• A. Closed ellipses
• B. Spirals towards the origin
• C. Straight lines away from the origin
• D. Spirals away from the origin

3. A system has eigenvalues $\pm3i$. Describe the trajectories.

• A. Closed ellipses around the origin
• B. Spirals towards the origin
• C. Straight lines through the origin
• D. A saddle point

4. $M$ has eigenvalue $-1$ with eigenvector $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$. If the system starts at $(2,2)$, what is $x(t)$?

• A. $2e^{t}$
• B. $2e^{-t}$
• C. $e^{-2t}$
• D. $2+e^{-t}$

5. Both eigenvalues are $-2$ and $-5$. What happens to $x$ and $y$ as $t$ increases?

• A. Both tend to zero
• B. Both grow without bound
• C. They oscillate with constant amplitude
• D. One grows and one decays