MAT244--2018F > Term Test 2
TT2-P3
Victor Ivrii:
(a) Find the general solution of
$$
\mathbf{x}'=\begin{pmatrix}\hphantom{-}2 &\hphantom{-}1\\
-3 &-2\end{pmatrix}\mathbf{x}.$$
(b) Sketch corresponding trajectories. Describe the picture (stable/unstable, node, focus, center, saddle).
(c) Solve
$$
\mathbf{x}'=\begin{pmatrix}\hphantom{-}2 &\hphantom{-}1\\
-3 &-2\end{pmatrix}\mathbf{x} +
\begin{pmatrix}\hphantom{-} \frac{4}{e^t+e^{-t}} \\
-\frac{12}{e^t+e^{-t}}\end{pmatrix},\qquad
\mathbf{x}(0)=\begin{pmatrix} 0 \\
0\end{pmatrix}.
$$
Boyu Zheng:
here is my answer
Zhanhao Ye:
The attachment is my solution.
Jingze Wang:
a)First, try to find the eigenvalues with respect to the parameter
$A=\begin{bmatrix}
2&1\\
-3&-2\\
\end{bmatrix}$
$det(A-rI)=(2-r)/(-2-r)+3=0$
$r^2-1=0$
$r=\pm1$
When r=-1, $3x_1+x_2=0$
We get $\begin{pmatrix}\hphantom{-} {1 }\\{-3}\end{pmatrix}$ is the corresponding eigenvector
When r=1, $x_1+x_2=0$
We get $\begin{pmatrix}\hphantom{-} {1 }\\{-1}\end{pmatrix}$ is the corresponding eigenvector
Then the general solution is $$y=c_1\begin{pmatrix}\hphantom{-} {1 }\\{-3}\end{pmatrix}e^{-t}+c_2\begin{pmatrix}\hphantom{-} {1 }\\{-1}\end{pmatrix}e^{t}$$
Michael Poon:
Seems like no one has added a phase portrait yet.
I attached it below, it is a saddle, with eigenvalues real and opposite.
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