Chapter 3.3, Problem 21P

Phase Portraits and Component Plots. In each of Problems $17$ through $24$, the eigenvalues and eigenvectors of a matrix $\text{A}$ are given. Consider the corresponding system $\text{x'}\text{=}\text{Ax}$. Without using a computer, draw each of the following graphs:

(a) Sketch a phase portrait of the system.

(b) Sketch the trajectory passing through the initial point $\left(2,3\right)$.

(c) For the trajectory in part (b), sketch the component plots of $x_1$ versus $t$ and of $x_2$ versus $t$ on the same set of axes.

${\lambda }_1=0.5,{\text{v}}_{\text{1}}=\left(\begin{array}{c}1\\ 4\end{array}\right);$

${\lambda }_2=-0.5,{\text{v}}_2=\left(\begin{array}{c}4\\ 1\end{array}\right)$