The state equation of a second-order linear system is given by
$$\dot{x}(t)=Ax(t), \:\:\:\:\:\:\:\:x(0)=x_{0}$$
For $x_{0}= \begin{bmatrix} 1\\ -1 \end{bmatrix},$ $x(t)= \begin{bmatrix} e^{-t}\\ -e^{-t} \end{bmatrix},$ and for $x_{0}= \begin{bmatrix} 0\\ 1 \end{bmatrix},$ $x(t) \begin{bmatrix} e^{-t}-e^{-2t}\\ -e^{-t}+2e^{-2t} \end{bmatrix}$. When $x_{0} = \begin{bmatrix} 3\\ 5 \end{bmatrix}$, $x(t)$ is
- $\begin{bmatrix} -8e^{-t}+11e^{-2t}\\ 8e^{-t}-22e^{-2t} \end{bmatrix} \\$
- $\begin{bmatrix} 11e^{-t}-8e^{-2t}\\ -11e^{-t}+16e^{-2t} \end{bmatrix} \\$
- $\begin{bmatrix} 3e^{-t}-5e^{-2t}\\ -3e^{-t}+10e^{-2t} \end{bmatrix} \\$
- $\begin{bmatrix} 5e^{-t}-3e^{-2t}\\ -5e^{-t}+6e^{-2t} \end{bmatrix}$