### Author Topic: Q6--T0401  (Read 2456 times)

#### Victor Ivrii

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##### Q6--T0401
« on: March 16, 2018, 08:11:48 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 $t\to \infty$.
$$\mathbf{x}' =\begin{pmatrix} 3 &-2\\ 2 &-2 \end{pmatrix}\mathbf{x}$$

#### Ge Shi

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##### Re: Q6--T0401
« Reply #1 on: March 16, 2018, 11:48:09 PM »
(a)
https://imgur.com/a/W9njS

(b)

When t approaches to infinity:
if C2 is not equal to zero ,the solution is unbounded.
if C2 is equal to zero, the solution approaches to zero.

Since $\lambda_1=-1$ , $\lambda_2=2$
Eigenvalues are real but unequal and have the opposite signs, x=0 is a saddle point and unstable.
I've attached the graph.
« Last Edit: March 17, 2018, 12:49:50 PM by Ge Shi »

#### Victor Ivrii

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##### Re: Q6--T0401
« Reply #2 on: March 17, 2018, 05:12:02 AM »
See my comment to your other post. And do not try to cover the same quiz in other sections!

#### Ge Shi

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##### Re: Q6--T0401
« Reply #3 on: March 17, 2018, 01:50:00 PM »
(a)