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Moderate

General Solutions of Linear Systems with Complex Eigenvalues

DIFFEQ-GV9WBG

Find the solution to the following linear system:

$$\dot{\vec{u}}=A\vec{u}$$

...where $A=\left[\begin{array}{rr} 1&2\\\ -2&1\end{array}\right]$.

A

$$c e^{(1+2i)t} \left[\begin{array}{c} 1 \\\ -i \end{array}\right]$$

B

$$y=c_1 e^{2t} \left[\begin{array}{c} \cos (2t) \\\ - \sin(2t)\end{array}\right] + c_2 e^{2t} \left[\begin{array}{c} \sin (2t) \\\ \cos(2t)\end{array}\right]$$

C

$$c_1 e^t \left[\begin{array}{c} \cos (2t) \\\ - \sin(2t)\end{array}\right] + c_2 e^t \left[\begin{array}{c} \sin (2t) \\\ \cos(2t)\end{array}\right]$$

D

$$c_1 e^t \left[\begin{array}{c} \cos (2t) \\\ \sin(2t)\end{array}\right] + c_2 e^t \left[\begin{array}{c} \sin (2t) \\\ \cos(2t)\end{array}\right]$$