Toronto Math Forum
MAT244-2018S => MAT244--Tests => Quiz-6 => Topic started by: Victor Ivrii on March 16, 2018, 08:14:20 PM
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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 $t\to \infty$.
$$\mathbf{x}' =\begin{pmatrix}
4 &-3\\
8 &-6
\end{pmatrix}\mathbf{x}$$
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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.
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Calculations are correct, but the phase portrait (extremely simple) is not. Where have you found this crapware?
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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?
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And how you enter matrix?
I put several in outlines. My favourite is pplane
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By adjusting the sliders.
I have edited my post to include the plot using pplane.
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You need also to include lines. Just make random clicks on the applet's field
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I added some lines to it now.