Author Topic: Q6--T0601  (Read 1889 times)

Victor Ivrii

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Q6--T0601
« on: March 16, 2018, 08:14:20 PM »
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

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Re: Q6--T0601
« Reply #1 on: March 17, 2018, 01:03:55 PM »
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.
« Last Edit: March 17, 2018, 08:34:02 PM by Mark Buchanan »

Victor Ivrii

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Re: Q6--T0601
« Reply #2 on: March 17, 2018, 01:30:25 PM »
Calculations are correct, but the phase portrait (extremely simple) is not. Where have you found this crapware?

Mark Buchanan

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Re: Q6--T0601
« Reply #3 on: March 17, 2018, 01:53:44 PM »
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

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Re: Q6--T0601
« Reply #4 on: March 17, 2018, 02:32:59 PM »
And how you enter matrix?

I put several in outlines. My favourite is pplane

Mark Buchanan

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Re: Q6--T0601
« Reply #5 on: March 17, 2018, 06:18:46 PM »
By adjusting the sliders.

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

Victor Ivrii

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Re: Q6--T0601
« Reply #6 on: March 17, 2018, 07:36:00 PM »
You need also to include lines. Just make random clicks on the applet's field


Mark Buchanan

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Re: Q6--T0601
« Reply #7 on: March 17, 2018, 08:37:00 PM »
I added some lines to it now.