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Differential Equations

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Linear System with Complex Eigenvalue: a Simple IVP

DIFFEQ-OFKAJL

Consider the IVP $x'=\left(\begin{array}{cc}1&-6\\\ 1&-3\end{array}\right)x$, $x(0)=\left(\begin{array}{c}2\\\ 1\end{array}\right)$.

The solution is

A

$e^{-t}\left(\begin{array}{c}2\cos\sqrt{2}t-\sqrt{2}\sin\sqrt{2}t\\\ \cos\sqrt{2}t\end{array}\right)+e^{-t}\left(\begin{array}{c}2\sin\sqrt{2}t+\sqrt{2}\cos\sqrt{2}t\\\ \sin\sqrt{2}t\end{array}\right)$

B

$e^{-t}\left(\begin{array}{c}2\cos\sqrt{2}t-\sqrt{2}\sin\sqrt{2}t\\\ \cos\sqrt{2}t\end{array}\right)$

C

$e^{-t}\left(\begin{array}{c}2\sin\sqrt{2}t+\sqrt{2}\cos\sqrt{2}t\\\ \sin\sqrt{2}t\end{array}\right)$

D

$e^{-t}\left(\begin{array}{c}2\cos\sqrt{2}t\\\ \cos\sqrt{2}t\end{array}\right)$