##### 47. The state equation of a second-order linea...

47. The state equation of a second-order linear system is given by

$\dot x(t)=Ax(t),x(0)=x_0$

For $x_0={[{1\over {-1}}]}$, $x(t)= \begin{bmatrix}e^{-t}\\-e^{-t}\end{bmatrix}$ 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

(A) $\begin{bmatrix}-8e^{-t}+11e^{-2t}\\8e^{-t}-22e^{-2t}\end{bmatrix}$

(B) $\begin{bmatrix}11e^{-t}-8e^{-2t}\\-11e^{-t}+16e^{-2t}\end{bmatrix}$

(C) $\begin{bmatrix}3e^{-t}-5e^{-2t}\\-3e^{-t}+10e^{-2t}\end{bmatrix}$

(D) $\begin{bmatrix}5e^{-t}-3e^{-2t}\\-5e^{-t}+6e^{-2t}\end{bmatrix}$

Hint:

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