The critical point $( 0,0 )$ of the system $\frac { d x } { d t } = - 6 x - 5 y , \frac { d y } { d t } = 3 x + 2 y$ is Select all that apply.
A) asymptotically stable
B) stable but not asymptotically stable
C) unstable
D) an attractor
E) a repeller

Question

Quizplus · Accepted Answer

The stability of a critical point can be determined by analyzing the eigenvalues of the Jacobian matrix of the system at that point. For the given system, the Jacobian matrix at $(0,0)$ is $\begin{pmatrix} -6 & -5 \ 3 & 2 \end{pmatrix}$. The eigenvalues of this matrix determine the behavior of the system near the critical point. Calculating the eigenvalues, we find they are both negative (real parts are negative), indicating that the critical point is asymptotically stable, meaning solutions that start near the critical point will tend towards it as $t \rightarrow \infty$, which also makes it an attractor.

The Critical Point (0,0)( 0,0 )(0,0) Of the System dxdt=−6x−5y,dydt=3x+2y\frac { d x } { d t } = - 6 x - 5 y , \frac { d y } { d t } = 3 x + 2 ydtdx​=−6x−5y,dtdy​=3x+2y

The Critical Point $( 0,0 )$ Of the System $\frac { d x } { d t } = - 6 x - 5 y , \frac { d y } { d t } = 3 x + 2 y$