Chapter 3.7, Problem 4E

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 $a$

4. $\frac{dY}{dt}=\left(\begin{array}{ll}a\hfill & a\hfill \\ 1\hfill & 0\hfill \end{array}\right)Y$