### Author Topic: Quiz6 LEC5101  (Read 1550 times)

#### Wang Jingyao

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##### Quiz6 LEC5101
« on: December 03, 2019, 10:53:51 PM »
a. Express the general solution of the given system of equations in terms of real-valued functions.
b. Also draw a direction field, sketch a few of the trajectories, and describe the behavior of
the solutions as $t \rightarrow \infty$.

$x’=$$\left ( \begin{matrix} 1 & 2 \\ -5 & -1 \end{matrix} \right )x det(A-\lambda I)=$$det \left [ \begin{matrix} 1-\lambda & 2 \\ -5 & -1-\lambda \end{matrix} \right ]=(\lambda-1)^2+10$

Solve for

$(\lambda-1)^2+10=0$

$\lambda_1=3i \;,\; \lambda_2=-3i$

Consider $\lambda=3i$

$x’=$$\left [ \begin{matrix} 1-3i & 2 \\ -5 & -1-3i \end{matrix} \right ] \left [ \begin{matrix} x_1 \\ x_2 \end{matrix} \right ] = \left [ \begin{matrix} 0 \\ 0 \end{matrix} \right ] let x_2=t, \left [ \begin{matrix} x_1 \\ x_2 \end{matrix} \right ] = \left [ \begin{matrix} -2 \\ 1-3i \end{matrix} \right ]t Consider e^{it}\left [ \begin{matrix} -2 \\ 1-3i \end{matrix} \right ] = (\cos3t+i\sin3t) \left [ \begin{matrix} -2 \\ 1-3i \end{matrix} \right ] = \left[ \begin{matrix} -2\cos3t\\ \cos3t+3\sin3t \end{matrix} \right] +i\left[ \begin{matrix} -2\sin3t\\ \sin3t-3\cos3t \end{matrix} \right] Therefore, x(t)=$$c_1 \left[ \begin{matrix} -2\cos3t\\ \cos3t+3\sin3t \end{matrix} \right] +c_2\left[ \begin{matrix} -2\sin3t\\ \sin3t-3\cos3t \end{matrix} \right]$