The geometric configuration of the solutions of $\frac { d x } { d t } = - 6 x - 5 y , \frac { d y } { d t } = 4 x + 2 y$ in the phase plane is
A) stable node
B) unstable node
C) stable spiral point
D) unstable spiral point
E) saddle point

Question

Quizplus · Accepted Answer

The given system of differential equations can be represented in matrix form as $\frac{d}{dt} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} -6 & -5 \ 4 & 2 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix}$. The eigenvalues of the matrix determine the nature of the phase plane. For this matrix, the eigenvalues are complex with negative real parts, indicating a stable spiral point.

The Geometric Configuration of the Solutions Of dxdt=−6x−5y,dydt=4x+2y\frac { d x } { d t } = - 6 x - 5 y , \frac { d y } { d t } = 4 x + 2 ydtdx​=−6x−5y,dtdy​=4x+2y

The Geometric Configuration of the Solutions Of $\frac { d x } { d t } = - 6 x - 5 y , \frac { d y } { d t } = 4 x + 2 y$