Chapter 3.7, Problem 6E

In Exercises 2-7, we consider the one-parameter families of linear systems depending on the parameter a. Each family therefore determines a curve in the trace-determinant plane. For each family,

(a) sketch the corresponding curve in the trace-determinant plane;
(b) in a brief essay, discuss different types of behaviors exhibited by the system as a increases along the real line (unless otherwise noted); and
(c) identify the values of a where the type of the system changes. These are the bifurcation values of a
6. $\frac{d\text{Y}}{dt}=\left(\begin{array}{rr}\hfill 2& \hfill 0\\ \hfill a& \hfill -3\end{array}\right)\text{Y}$