MAT244-2018S > Quiz-6

Q6--T0601

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

Mark Buchanan:
a)

First we find the eigenvalues

$$det\begin{bmatrix}4-r & -3\\8 & -6-r\end{bmatrix} = (r-4)(r+6)-24 = (r^2+2r) = r(r+2)$$

$$r_1 = 0, r_2 = -2$$

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/Ch7-LN9.pdf)

The plot approaches the vector $\begin{bmatrix}3\\4\end{bmatrix}$ as t approaches infinity.

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

Mark Buchanan:
I got it from here: http://mathlets.org/mathlets/linear-phase-portraits-matrix-entry/

I couldn't find one that worked better.  Do you have any suggestions of what to use?

Victor Ivrii:
And how you enter matrix?

I put several in outlines. My favourite is pplane