# Toronto Math Forum

## MAT244--2018F => MAT244--Tests => Term Test 2 => Topic started by: Victor Ivrii on November 20, 2018, 05:52:12 AM

Title: TT2A-P3
Post 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
$$\mathbf{x}'=\begin{pmatrix}\hphantom{-}4 & \ 1\\ -3 &0\end{pmatrix}\mathbf{x} + \begin{pmatrix} \hphantom{-}\frac{4e^{4t}}{e^t+1} \\ -\frac{4e^{4t}}{e^t+1}\end{pmatrix},\qquad \mathbf{x}(0)=\begin{pmatrix}-1 \\ \hphantom{-}3\end{pmatrix}.$$
Title: Re: TT2A-P3
Post by: Mallory Schneider on November 20, 2018, 12:08:17 PM
Part a) and b)
Title: Re: TT2A-P3
Post by: Michael Poon on November 20, 2018, 12:17:52 PM
I think you could also characterise the phase portrait as a node? (unstable node)
Title: Re: TT2A-P3
Post by: Mallory Schneider on November 20, 2018, 12:27:36 PM
Part C
Title: Re: TT2A-P3
Post by: Victor Ivrii on November 25, 2018, 12:34:25 PM
I think you could also characterise the phase portrait as a node? (unstable node)
Indeed

Computer generated