Chapter 6.3, Problem 23E

To determine

To find: The general solution to the system of first order homogeneous equations $X^{\prime }=AX$ by using matrices $\varphi \left(t\right)=\left[\begin{array}{cc}2e^{4t}& e^{-t}\\ 3e^{4t}& -e^{-t}\end{array}\right]$ and $A=\left[\begin{array}{cc}1& 2\\ 3& 2\end{array}\right]$, find a solution that satisfies the initial conditions $\{\begin{array}{l}{x}^{\prime }=x+2y\\ y^{\prime }=3x+2y\\ x\left(0\right)=3,y\left(0\right)=2\end{array}$ and graph the components individually as functions of $t$ as well as parametrically.