TT2-P4

[Victor Ivrii]

(a) Find the general real solution to
$$
\mathbf{x}'=\begin{pmatrix}
\hphantom{-}5 & \hphantom{-}5\\
-5 &-1\end{pmatrix}\mathbf{x}.$$
(b) Sketch trajectories. Describe the picture (stable/unstable, node, focus, center, saddle).

[Jingze Wang]

First, try to find the eigenvalues with respect to the parameter


$A=\begin{bmatrix}
5&5\\
-5&-1\\
\end{bmatrix}$


$det(A-rI)=(5-r)(-1-r)+25=0$

$r^2-4r+20=0$

$r=\frac{4\pm\sqrt{-64}}{2}$

$r=2\pm4i$

Since they are complex conjugates
Then just use one of the eigenvector to find real solution
Use eigenvalue $r=3+4i$ to find its corresponding eigenvector

\begin{bmatrix}
3-4i&5\\
-5&-3-4i\\
\end{bmatrix}


The eigenvector is
$\begin{bmatrix}
5\\
4i-3
\end{bmatrix}$

$X=e^{2+4i}\begin{bmatrix}5\\4i-3\end{bmatrix}\cos4t+i\sin4t$


Rearrange this, we get $U=e^{2t}\begin{bmatrix}
\cos4t\\
-3cos4t-4\sin4t
\end{bmatrix}$

Also, $V=e^{2t}\begin{bmatrix}
5\sin4t\\
4\cos4t-3\sin4t
\end{bmatrix}$

And they are real valued solutions

Since -5<0, it is clockwise

Also real parts is 2>0, it is unstable spiral

[Jingze Wang]

Also, I am wondering can I just say it is spiral instead of focus

[Mengfan Zhu]

Hello, this is my answer.
To be clear, I did it step by step to get the general real solution ^_^

[Victor Ivrii]

Computer-generated