Toronto Math Forum
MAT2442018F => MAT244Tests => Term Test 2 => Topic started by: Victor Ivrii on November 20, 2018, 05:52:12 AM

(a) Find the general solution of
$$
\mathbf{x}'=\begin{pmatrix} \ 4 & \ 1\\
3 &0\end{pmatrix}\mathbf{x}.$$
(b) Sketch corresponding trajectories. Describe the picture (stable/unstable, node, focus, center, saddle).
(c) Solve
$$
\mathbf{x}'=\begin{pmatrix}\hphantom{}4 & \ 1\\
3 &0\end{pmatrix}\mathbf{x} +
\begin{pmatrix} \hphantom{}\frac{4e^{4t}}{e^t+1} \\
\frac{4e^{4t}}{e^t+1}\end{pmatrix},\qquad
\mathbf{x}(0)=\begin{pmatrix}1 \\
\hphantom{}3\end{pmatrix}.
$$

Part a) and b)

I think you could also characterise the phase portrait as a node? (unstable node)

Part C

I think you could also characterise the phase portrait as a node? (unstable node)
Indeed
Computer generated