Question

classify the critical point \\((0, 0)\\) of the given linear system by computing the trace \\(\tau\\) and determinant \\(\delta\\) and using the figure.

\\
\

$$\begin{aligned} x &= -5x + 6y \\\\ y &= 3x + 4y \\end{aligned}$$

\\

unstable spiral
degenerate unstable node
center
saddle
stable node
stable spiral
degenerate stable node
unstable node

Explanation:

Set up the coefficient matrix

$$ A =
LATEXBLOCK0
$$

Calculate the trace and determinant

$$
LATEXBLOCK1
$$

Classify the critical point

Since \(\Delta = -38 < 0\), the critical point \((0,0)\) is a saddle.

Answer:

• unstable spiral
• degenerate unstable node
• center
• saddle (Correct answer)
• stable node
• stable spiral
• degenerate stable node
• unstable node