# Toronto Math Forum

## MAT244-2018S => MAT244--Tests => Quiz-6 => Topic started by: Victor Ivrii on March 16, 2018, 08:11:48 PM

Title: Q6--T0401
Post by: Victor Ivrii 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 $t\to \infty$.
$$\mathbf{x}' =\begin{pmatrix} 3 &-2\\ 2 &-2 \end{pmatrix}\mathbf{x}$$
Title: Re: Q6--T0401
Post by: Ge Shi on March 16, 2018, 11:48:09 PM
(a)
https://imgur.com/a/W9njS (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.
Title: Re: Q6--T0401
Post by: Victor Ivrii 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!
Title: Re: Q6--T0401
Post by: Ge Shi on March 17, 2018, 01:50:00 PM
(a)