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


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