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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  

  1. $\begin{bmatrix} -8e^{-t}+11e^{-2t}\\ 8e^{-t}-22e^{-2t} \end{bmatrix} \\$
  2. $\begin{bmatrix} 11e^{-t}-8e^{-2t}\\ -11e^{-t}+16e^{-2t} \end{bmatrix} \\$
  3. $\begin{bmatrix} 3e^{-t}-5e^{-2t}\\ -3e^{-t}+10e^{-2t} \end{bmatrix} \\$
  4. $\begin{bmatrix} 5e^{-t}-3e^{-2t}\\ -5e^{-t}+6e^{-2t} \end{bmatrix}$
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