MAT244--2018F > Term Test 2

TT2A-P4

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Victor Ivrii:
(a) Find the general real solution to
$$\mathbf{x}'=\begin{pmatrix} \hphantom{-}1 & \hphantom{-}2\\ -5 &-1\end{pmatrix}\mathbf{x}.$$
(b) Sketch trajectories. Describe the picture (stable/unstable, node, focus, center, saddle).

Samarth Agarwal:
First, try to find the eigenvalues with respect to the parameter
$$A=\begin{bmatrix} 1&2\\ -5&-1\\ \end{bmatrix}$$
$$det(A-rI)=(1-r)(-1-r)+10=0$$
$$r^2 + 9 = 0$$
$$r = \pm 3i$$
The eigenvector is \\ \begin{bmatrix} -2\\ 1-3i \end{bmatrix}
Therefore x_1 =
$$\begin{bmatrix} -2\\ 1-3i \end{bmatrix} (\cos3t + i\sin3t)$$
$$= \begin{bmatrix} -2\cos3t \\ \cos3t + 3\sin3t \end{bmatrix} + i \begin{bmatrix} -2\sin3t\\ -3\cos3t + \sin3t \end{bmatrix}$$
Therefore the general solution
$$x(t) = c_1 \begin{bmatrix} -2\cos3t \\ \cos3t + 3\sin3t \end{bmatrix} + c_2 \begin{bmatrix} -2\sin3t\\ -3\cos3t + \sin3t \end{bmatrix}$$

Mengfan Zhu:
To be clear, I did it step by step to get the general real solution
If there are any mistakes, please tell me below ^_^

Jingze Wang:
Hello Samarth, I think your graph is not right, since the eigenvalues have no real parts, then graph should be center instead of spiral.

Michael Poon:
Yes, I agree it should be a centre (CW). Of course, with axes: $x_1$ horizontally and $x_2$ vertically.