MAT244--2018F > Final Exam

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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.