7.4 Basic theory systems of Linear ODE

Motivating example

$$\begin{array}{r}{\vec{x}}^{\prime }\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]\vec{x}⟹\begin{array}{c}{x}_1^{\prime }(t)=2x_1(t)-x_2(t)\\ x_2^{\prime }=3x_1(t)-2x_2(t)\end{array}\\ \text{this gives us two solutions}\\ \\ \text{a) show that the given vector functions are indeed solutions}\\ \left[\begin{array}{c}{e}^t\\ e^t\end{array}\right]?=\left[\begin{array}{c}\frac{2-1}{3-2}\end{array}\right]\left[\begin{array}{c}{e}^t\\ e^t\end{array}\right]⟹\text{ this works because it equals}\\ \text{ b) show that}\vec{x}(t)=c_1{\vec{x}}_1+c_2{\vec{x}}_2\text{ is also a solution for any consatnt }{c}_1,c_2\\ \left[\begin{array}{c}{c}_1{e}^t\\ c_1{e}^t\end{array}\right]+\left[\begin{array}{c}{c}_2{e}^t\\ 3c_2{e}^t\end{array}\right]=\end{array}$$

Phase plane