Chapter 3.4, Problem 20P

Dependence on a Parameter. In each of Problems $13$ and $20$, the coefficient matrix contains a parameter $\alpha$. In each of these problems:

(a) Determine the eigenvalues in terms of $\alpha$.

(b) Find the critical value or values of $\alpha$ where the qualitative nature of phase portrait for the system changes.

(c) Draw a phase portrait for a value of $\alpha$ slightly below, and for another value slightly above, each critical value.

$\text{x}\text{'}=\left(\begin{array}{cc}4& \alpha \\ 8& -6\end{array}\right)\text{x}$