# Section 7.6 – Complex Eigenvalues

Directions. The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it.
Note: The fundamental matrix mentioned below is discussed in Section 7.7.
1. Find the general solution of the system and the fundamental matrix. Classify the type of the critical point (0,0), and determine whether it is stable or unstable. Sketch the phase portrait.${\bf x}'=\begin{pmatrix} 1&-1\\5&-3\end{pmatrix}{\bf x}$

The general solution is \begin{align}\mathbf{x}(t)=&C_1e^{-t}\left[\begin{pmatrix}1\\2\\\end{pmatrix}\cos(t)+\begin{pmatrix}0\\1\end{pmatrix}\sin(t)\right]\\&+C_2e^{-t}\left[\begin{pmatrix}1\\2\end{pmatrix}\sin(t)+\begin{pmatrix}0\\-1\end{pmatrix}\cos(t)\right]\end{align} and the fundamental matrix is $\begin{pmatrix}e^{-t}\cos(t)&e^{-t}\sin(t)\\e^{-t}(2\cos(t)+\sin(t))&e^{-t}(2\sin(t)-\cos(t))\end{pmatrix}$ The critical point $$(0,0)$$ type is a spiral sink and asymptotically stable. Refer to the video for a sketch of the phase portrait.

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