GATE EC 2014 Set 3 Q47 — Linear Algebra

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