Exercise
Consider the system $ \dot x = Ax+bu$ with $$ A = \begin{bmatrix} -6 & 2 \\ -6 & 1 \end{bmatrix},~~~ b= \begin{bmatrix} 1 \\ 0 \end{bmatrix},~~~ c=\begin{bmatrix} 1 & 1 \end{bmatrix} $$
Calculate the state and output responses, $x(t)$ and $y(t)$, with inital values $x(0)= \begin{bmatrix} 2 & 1 \end{bmatrix}^T $
Solution
The homogeneous time response of the state is $$ x_{0}(t) = e^{At}x(0) = e^{At} \begin{bmatrix} 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 2\alpha_0 -10\alpha_1 \\ -11\alpha_1+\alpha_0 \end{bmatrix} = \begin{bmatrix} 6e^{-3t}-4 e^{-2t} \\ 9e^{-3t}-8e^{-2t} \end{bmatrix} $$
While that of the output is $$y_0(t)=15e^{-3t}-12e^{-2t} $$
Question
I do not understand how they jumped from $e^{At} \begin{bmatrix} 2 \\ 1 \end{bmatrix}$ to $\begin{bmatrix} 2\alpha_0 -10\alpha_1 \\ -11\alpha_1+\alpha_0 \end{bmatrix}$ and to $\begin{bmatrix} 6e^{-3t}-4 e^{-2t} \\ 9e^{-3t}-8e^{-2t} \end{bmatrix}$.
What is $x_0(t)$? How can one control the initial state when the to be regulated process has not started (t=0) yet? Is there a real life example?
How did they get to $y_0(t)$?
First answer is quite long, but eventually boils down to diagonalizing matrix A to a matrix in Jordan form. I can write it out, but I am not sure if you're still looking for the answer?
Short answer to the second question is $y = Cx$, therefore:
$$y_0(t) = Cx_0(t) = \begin{pmatrix}1 & 1\end{pmatrix}\begin{pmatrix}x_{0_1}(t) \\ x_{0_2}(t)\end{pmatrix}$$