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]
$
