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Differential Equations

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Identifying the Solution to the System of DEs 2

DIFFEQ-TUNRET

Consider the linear system:

$$X'=\left(\begin{array}{ccc} 3 & 2 &-3 \\\ 1 & 1 & 1 \\\ 0 & -4 & -4 \end{array}\right)X$$

...the eigenvalues of the coefficient matrix and one of its corresponding eigenvectors for each eigenvalue are given:

$$\lambda_{1}=-2+I, V_{1}=\left(\begin{array}{c} 3+i \\\ -2-i \\\ 4 \end{array}\right);\lambda_{2}=-2-I, V_{2}=\left(\begin{array}{c} 3-i \\\ -2+i \\\ 4 \end{array}\right);\lambda_{3}=4, V_{3}=\left(\begin{array}{c} -7 \\\ -2 \\\ 1 \end{array}\right)$$

The general solution is of the form:

A

$X=c_{1}\left(\left(\begin{array}{c} 3 \\\ -2 \\\ 4 \end{array}\right)\cos t+\left(\begin{array}{c} 1 \\\ -1 \\\ 0 \end{array}\right)\sin t\right) e^{-2t}+c_{2}\left(\left(\begin{array}{c} 3 \\\ -2 \\\ 4 \end{array}\right)\sin t+\left(\begin{array}{c} 1 \\\ -1 \\\ 0 \end{array}\right)\cos t\right) e^{-2t}+c_{3}\left(\begin{array} -7 \\\ -2 \\\ 1 \end{array}\right) e^{4t}$

B

$X=c_{1}\left(\left(\begin{array}{c} 3 \\\ -2 \\\ 4 \end{array}\right)\cos t-\left(\begin{array}{c} 1 \\\ -1 \\\ 0 \end{array}\right)\sin t\right) e^{-2t}+c_{2}\left(\left(\begin{array}{c} 3 \\\ -2 \\\ 4 \end{array}\right)\sin t-\left(\begin{array}{c} 1 \\\ -1 \\\ 0 \end{array}\right)\cos t\right) e^{-2t}+c_{3}\left(\begin{array} -7 \\\ -2 \\\ 1 \end{array}\right) e^{4t}$

C

$X=c_{1}\left(\left(\begin{array}{c} 3 \\\ -2 \\\ 4 \end{array}\right)\cos t+\left(\begin{array}{c} 1 \\\ -1 \\\ 0 \end{array}\right)\sin t\right) e^{-2t}+c_{2}\left(\left(\begin{array}{c} 3 \\\ -2 \\\ 4 \end{array}\right)\sin t-\left(\begin{array}{c} 1 \\\ -1 \\\ 0 \end{array}\right)\cos t\right) e^{-2t}+c_{3}\left(\begin{array} -7 \\\ -2 \\\ 1 \end{array}\right) e^{4t}$

D

$X=c_{1}\left(\left(\begin{array}{c} 3 \\\ -2 \\\ 4 \end{array}\right)\cos t-\left(\begin{array}{c} 1 \\\ -1 \\\ 0 \end{array}\right)\sin t\right) e^{-2t}+c_{2}\left(\left(\begin{array}{c} 3 \\\ -2 \\\ 4 \end{array}\right)\sin t+\left(\begin{array}{c} 1 \\\ -1 \\\ 0 \end{array}\right)\cos t\right) e^{-2t}+c_{3}\left(\begin{array} -7 \\\ -2 \\\ 1 \end{array}\right) e^{4t}$

E

None of the above