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Differential Equations

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Difficult

Finding Solution to the System of Differential Equations 6

DIFFEQ-H6JE1N

The general solution of the system:

$$X'=\left(\begin{array}{ccc}2 & 1 & 6 \\\ 0 & 2 & 5 \\\ 0 & 0 & 2\end{array}\right)X$$

...is:

A

$X=c_{1}\left(\begin{array}{c}1 \\\ 0 \\\ 0\end{array}\right)e^{2t}+c_{2}\left(\begin{array}{c}1 \\\ 0 \\\ 0\end{array}\right)te^{2t}+c_{3}\left(\begin{array}{c}1 \\\ 0 \\\ 0\end{array}\right)t^{2}e^{2t}$

B

$X=c_{1}\left(\begin{array}{c}0 \\\ 1 \\\ 0\end{array}\right)e^{2t}+c_{2}\left(\begin{array}{c}0 \\\ 1 \\\ 1\end{array}\right)te^{2t}+c_{3}\left(\begin{array}{c}0 \\\ 1 \\\ 0\end{array}\right)t^{2}e^{2t}$

C

$X=c_{1}\left(\begin{array}{c}1 \\\ 0 \\\ 0\end{array}\right)e^{2t}+c_{2}\left[\left(\begin{array}{c}1 \\\ 0 \\\ 0\end{array}\right)te^{2t}+\left(\begin{array}{c}0 \\\ 1 \\\ 0 \end{array}\right)e^{2t}\right]+c_{3}\left(\begin{array}{c}1 \\\ 0 \\\ 0\end{array}\right)t^{2}e^{2t}$

D

$X=c_{1}\left(\begin{array}{c}1 \\\ 0 \\\ 0\end{array}\right)e^{2t}+c_{2}\left[\left(\begin{array}{c}1 \\\ 0 \\\ 0\end{array}\right)te^{2t}+\left(\begin{array}{c}0 \\\ 1 \\\ 0 \end{array}\right)e^{2t}\right]+c_{3}\left[\left(\begin{array}{c}1 \\\ 0 \\\ 0\end{array}\right)t^{2}e^{2t}+\left(\begin{array}{c}0 \\\ 1 \\\ 0 \end{array}\right)te^{2t}+\left(\begin{array}{c}0 \\\ -\cfrac{6}{5} \\\ \cfrac{1}{5}\end{array}\right)e^{2t}\right]$

E

None of the above