Toronto Math Forum
MAT2442018S => MAT244Tests => Quiz6 => Topic started 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 realvalued functions.
b. Also draw a direction field, sketch a few of the trajectories, and describe the behavior of the solutions as $t\to \infty$.
$$\mathbf{x}' =\begin{pmatrix}
3 &2\\
2 &2
\end{pmatrix}\mathbf{x}$$

(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.

See my comment to your other post. And do not try to cover the same quiz in other sections!

(a)