MAT244--2018F > Final Exam
FE-P5
Victor Ivrii:
Typed solutions only. Upload only pictures (at all stationary points on one picture and a general phase portrait on another; for general one can use computer generated)
For the system of ODEs
\begin{equation*}
\left\{\begin{aligned}
&x' = x(3x +2y -30)\, , \\
&y' = y(2y-x-6)\,.
\end{aligned}\right.
\end{equation*}
(a) Describe the locations of all critical points.
(b) Classify their types (including whatever relevant: stability, orientation, etc.).
(c) Sketch the phase portraits near the critical points.
(d) Sketch the full phase portrait of this system of ODEs.
Hint: avoid redundancy: asymptotically (un)stable node, unstable node, stable center
Doris Zhuomin Jia:
a) 𝑥(3𝑥+2𝑦−30)=0,𝑦(2𝑦−𝑥−6)=0
The critical points are (0,0), (0,3), (10,0) and (6,6)
Jerry Qinghui Yu:
$$J=\begin{bmatrix}
6x+2y-30 & 2x\\
-y & 4y-x-6
\end{bmatrix}$$
at (0,0):
$$J=\begin{bmatrix}
-30 & 0\\
0 & -6
\end{bmatrix}$$
diagonal matrix with negative eigenvalues => stable node
at (0,3):
$$J=\begin{bmatrix}
-24 & 0\\
-3 & 6
\end{bmatrix}$$
triangular matrix with eigenvalues -24 and 6 => saddle
at (10,0):
$$J=\begin{bmatrix}
30 & 20\\
0 & -16
\end{bmatrix}$$
diagonal matrix with eigenvalues 30 and -16 => saddle
at (6,6):
$$J=\begin{bmatrix}
18 & 12\\
-6 & 12
\end{bmatrix}$$
eigenvalues are $15+3i\sqrt{7}, 15-3i\sqrt{7}$ => unstable spiral
Yvette Yu:
Here is diagram for (c) and (d)
Jingze Wang:
This is the computer generated global phase portrait.
We already know that Wolfram Alpha provides rather crappy pictures here. V.I.
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