Q6--T0401

[Victor Ivrii]

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]

(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.

[Victor Ivrii]

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

[Ge Shi]

(a)