?

Differential Equations

Free Version

Upgrade subject to access all content

Moderate

Solution to System Involving Complex Eigenvalues

DIFFEQ-LYVRTX

Find the general solution to the system $X'(t)=AX(t)$, where $A=\left( \begin{array}{cc} 3 & -2 \\\ 4 & -1 \end{array}\right)$.

A

$X=c_{1}\left[\left(\begin{array}{c} 1 \\\ 1\end{array}\right)\cos 2t+\left(\begin{array}{c} 0 \\\ -1\end{array}\right)\sin 2t\right] e^{t}+c_{2}\left[\left(\begin{array}{c} 1 \\\ 1\end{array}\right)\sin 2t-\left(\begin{array}{c} 0 \\\ -1\end{array}\right)\cos 2t\right] e^{t}$

B

$X=c_{1}\left[\left(\begin{array}{c} 1 \\\ 1\end{array}\right)\sin 2t+\left(\begin{array}{c} 0 \\\ -1\end{array}\right)\cos 2t\right] e^{t}+c_{2}\left[\left(\begin{array}{c} 1 \\\ 1\end{array}\right)\sin 2t+\left(\begin{array}{c} 0 \\\ -1\end{array}\right)\cos 2t\right] e^{t}$

C

$X=c_{1}\left[\left(\begin{array}{c} 1 \\\ 1\end{array}\right)\cos 2t-\left(\begin{array}{c} 0 \\\ -1\end{array}\right)\sin 2t\right] e^{t}+c_{2}\left[\left(\begin{array}{c} 1 \\\ 1\end{array}\right)\sin 2t+\left(\begin{array}{c} 0 \\\ -1\end{array}\right)\cos 2t\right] e^{t}$

D

$X=c_{1}\left[\left(\begin{array}{c} 1 \\\ 1\end{array}\right)\cos 2t-\left(\begin{array}{c} 0 \\\ -1\end{array}\right)\sin 2t\right] e^{t}+c_{2}\left[\left(\begin{array}{c} 1 \\\ 1\end{array}\right)\cos 2t+\left(\begin{array}{c} 0 \\\ -1\end{array}\right)\sin 2t\right] e^{t}$

E

None of the above