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Differential Equations

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Moderate

Nonhomogeneous Linear System with Complex Eigenvalues

DIFFEQ-1XJPXB

Consider the system of equations:

$$ x'=Ax+\left(\begin{array}{c} -2\\\ 1\end{array}\right)=\left(\begin{array}{cc} 2&-5\\\ 1&-2\end{array}\right)x+ \left(\begin{array}{c} -2\\\ 1\end{array}\right) $$

Which one of the following statements is wrong?

A

One solution is given by: $ x=\left(\begin{array}{cc} 2&-5\\\ 1&-2\end{array}\right)^{-1}\left(\begin{array}{c} 2\\\ -1\end{array}\right)=\left(\begin{array}{c} -9\\\ -4\end{array}\right). $

B

Any solution of the system can be written as $x=x_h+\left(\begin{array}{c} -9\\\ -4\end{array}\right)$, where $x_h$ solves the homogeneous system $x'=Ax$.

C

The general solution is given by
$ x(t)=C_1\left(\begin{array}{c} 2\cos t-\sin t\\\ \cos t\end{array}\right) +C_2\left(\begin{array}{c} 2\sin t+\cos t\\\ \sin t\end{array}\right)+\left(\begin{array}{c} -9\\\ -4\end{array}\right). $

D

The general solution is given by $ x(t)=C_1\left(\begin{array}{c} 2\cos t-\sin t\\\ \cos t\end{array}\right) +C_2\left(\begin{array}{c} 2\sin t+\cos t\\\ \sin t\end{array}\right)+C_3\left(\begin{array}{c} -9\\\ -4\end{array}\right). $