Chapter 9.1, Problem 54E

Consider a door that opens to only one side (as most doors do). A spring mechanism closes the door automatically. The state of the door at a given time t (measured in seconds) is determined by the angular displacement $\theta (t)$ (measured in radians) and the angular velocity $\omega (t)=d\theta /dt$. Note that θ is always positive or zero (since the door opens to only one side), but $\omega$ can be positive or negative (depending on whether the door is opening or closing).

When the door is moving freely (nobody is pushing or pulling), its movement is subject to the following differential equations:
$\left|\begin{array}{c}\frac{d\theta }{dt}=\omega \\ \frac{d\omega }{dt}=-2\theta -3\omega \end{array}\right|\begin{array}{c}(thedefinitionof\omega )\\ (-2\theta reflectstheforceofthespring,and-3\omega \mathrm{mod}elsfriction)\end{array}$

a. Sketch a phase portrait for this system.
b. Discuss the movement of the door represented by the qualitatively different trajectories. For which initial states does the door slam (i.e., reach $\theta =0$ with velocity $\omega <0$ )?