The critical point $( 0,0 )$ of the system $\frac { d x } { d t } = 5 x + y , \frac { d y } { d t } = 3 x + 3 y$ 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 system's Jacobian matrix at that point. For the given system, the Jacobian matrix is $\begin{pmatrix} 5 & 1 \ 3 & 3 \end{pmatrix}$. The eigenvalues of this matrix are found by solving the characteristic equation $\det(J - \lambda I) = 0$, where $J$ is the Jacobian matrix and $I$ is the identity matrix. This leads to the equation $(5 - \lambda)(3 - \lambda) - 3 = \lambda^2 - 8\lambda + 12 = 0$. Solving this quadratic equation for $\lambda$ gives two eigenvalues. Without explicitly solving it here, if both eigenvalues have positive real parts, the critical point is unstable and acts as a repeller. Given the coefficients of the system, it's clear that the real parts of the eigenvalues are positive (since the trace of the matrix, which is the sum of the eigenvalues, is positive and greater than the determinant, which implies that the eigenvalues themselves must be positive), indicating that the system is indeed unstable and acts as a repeller.

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

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