I am a bit confused about the following from my lecture note:
Suppose we have
\begin{equation} \begin{aligned} \frac{dx}{dt}&=-\epsilon y \\ \frac{dy}{dt}&=by+cx \end{aligned} \end{equation}
How do we get $$x(t)=A\exp\big(\frac{\epsilon c}{b}t\big)+B\exp\big(-|b|t\big)$$
I know the following
\begin{equation} \begin{aligned} \begin{bmatrix}\dot{x}\\ \dot{y}\end{bmatrix}=\begin{bmatrix}0 & -\epsilon\\ c & b\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix} \end{aligned} \end{equation}Since the matrix is upper-triangular matrix, so I have the following:
$$\begin{bmatrix}x\\ y\end{bmatrix} = \exp(\begin{bmatrix}0 & -\epsilon\\ c & b\end{bmatrix})\begin{bmatrix}x_0\\ y_0\end{bmatrix} $$
Then how do I go to the next step? Please advise, thanks!