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Differential Equations

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Moderate

Linear System with Complex Eigenvalues: a $3\times 3$ System

DIFFEQ-0ZOH13

Which one is the general solution of the linear system?

$$ x'=\left( \begin{array}{ccc} 1&0&0\\\ 2&1&-1\\\ 3&2&1 \end{array}\right)x $$

A

$x(t)=C_1e^t\left(\begin{array}{c}2\\\ -3\\\ 4\end{array}\right) +C_2e^t\left(\begin{array}{c}0\\\ \sin\sqrt{2}t\\\ \sqrt{2}\cos\sqrt{2} t\end{array}\right) +C_3e^t\left(\begin{array}{c}0\\\ \cos\sqrt{2}t\\\ -\sqrt{2}\sin\sqrt{2} t\end{array}\right).$

B

$x(t)=C_1e^t\left(\begin{array}{c}2\\\ -3\\\ 4\end{array}\right) +C_2e^{(1+\sqrt{2}i)t}\left(\begin{array}{c}0\\\ 1\\\ -\sqrt{2}i\end{array}\right)$.

C

$x(t)=C_1e^t\left(\begin{array}{c}2\\\ -3\\\ 4\end{array}\right) +C_2e^t\left(\begin{array}{c}0\\\ \cos\sqrt{2}t\\\ \sin\sqrt{2} t\end{array}\right) +C_3e^t\left(\begin{array}{c}0\\\ \sin\sqrt{2}t\\\ -\cos\sqrt{2} t\end{array}\right).$

D

$x(t)=C_1e^t\left(\begin{array}{c}2\\\ -3\\\ 4\end{array}\right) +C_2e^t\left(\begin{array}{c}0\\\ \cos\sqrt{2}t\\\ \sqrt{2}\sin\sqrt{2} t\end{array}\right) +C_3e^t\left(\begin{array}{c}0\\\ \sin\sqrt{2}t\\\ -\sqrt{2}\cos\sqrt{2} t\end{array}\right).$