Toronto Math Forum
MAT2442018S => MAT244Tests => Quiz6 => Topic started by: Victor Ivrii on March 16, 2018, 08:14:20 PM

a. Express the general solution of the given system of equations in terms of realvalued functions.
b. Also draw a direction field, sketch a few of the trajectories, and describe the behavior of
the solutions as $t\to \infty$.
$$\mathbf{x}' =\begin{pmatrix}
4 &3\\
8 &6
\end{pmatrix}\mathbf{x}$$

a)
First we find the eigenvalues
$$det\begin{bmatrix}4r & 3\\8 & 6r\end{bmatrix} = (r4)(r+6)24 = (r^2+2r) = r(r+2)$$
$$r_1 = 0, r_2 = 2$$
The associated eigenvector for $r_1$ is:
$$r_1=0: Null\begin{bmatrix}4 & 3\\8 & 6\end{bmatrix} = Null\begin{bmatrix}4 & 3\\0 & 0\end{bmatrix} \implies 4\xi_1 = 3\xi_2 \implies \xi^{(1)} = \begin{bmatrix}3\\4\end{bmatrix} $$
The associated eigenvector for $r_2$ is:
$$r_2=2: Null\begin{bmatrix}6 & 3\\8 & 4\end{bmatrix} = Null\begin{bmatrix}2 & 1\\0 & 0\end{bmatrix} \implies 2\xi_1 = \xi_2 \implies \xi^{(2)} = \begin{bmatrix}1\\2\end{bmatrix} $$
This gives us our general solution:
$$X(t) = c_1\begin{bmatrix}3\\4\end{bmatrix} + c_2e^{2t}\begin{bmatrix}1\\2\end{bmatrix}$$
b)
The plot follows the same idea as 1. e) in this handout: (www.math.toronto.edu/courses/mat244h1/20181/LN/Ch7LN9.pdf)
The plot approaches the vector $\begin{bmatrix}3\\4\end{bmatrix}$ as t approaches infinity.

Calculations are correct, but the phase portrait (extremely simple) is not. Where have you found this crapware?

I got it from here: http://mathlets.org/mathlets/linearphaseportraitsmatrixentry/
I couldn't find one that worked better. Do you have any suggestions of what to use?

And how you enter matrix?
I put several in outlines. My favourite is pplane

By adjusting the sliders.
I have edited my post to include the plot using pplane.

You need also to include lines. Just make random clicks on the applet's field

I added some lines to it now.