Fizaa A.

asked • 07/30/20

Please help with the following question

If one eigenvalue of a 2 by 2 matrix LaTeX: E_{2 \times2}  is  LaTeX: \lambda = {\vec i}   and a corresponding eigenvector of LaTeX: {\vec i}  is

LaTeX: {\vec v} = \left(\begin{array}{r}
1\\
\\
{\vec i}
\end{array}\right)   , then the general solution of the system ODEs:

LaTeX: {\vec x^{\prime} (t)} = E_{2 \times 2} \cdot {\vec x(t)}


A.)is  LaTeX: {\vec x(t)} = C_1 \cdot \left[\begin{array}{c}
\cos t\\
\\
\sin t
\end{array}\right] \, + \, C_2 \cdot \left[\begin{array}{c}
\sin t\\
\\
\cos t
\end{array}\right]

B.) can't be determined. Solving the system requires additional information.

C.)is LaTeX: {\vec x(t)} = \left[\begin{array}{c}
\sin t \\
\\
\cos t
\end{array}\right]
\, + \, C_2 \, \left[\begin{array}{c}
\cos t\\
\\
\sin t
\end{array}
\right]

D.) is LaTeX: {\vec x(t)} = C_1 \cdot \left[\begin{array}{c}
\cos t\\
\\
- \, \sin t
\end{array}\right] + C_2 \, \left[\begin{array}{c}
\sin t \\
\\
\cos t
\end{array} \right]


I think its either C or D. 

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