determine minimal conditions on the real constant $\mu$ so that $(0, 0)$ is a center for the linear system
\
\begin{aligned}
x &= -\mu x + y \\
y &= -x + \mu y.
\end{aligned}
\
$|\mu|$ ?

Question

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Linear Systems of ODEs", "Stability of Critical Points" ], "current_concepts": [ "Linear Systems of ODEs", "Stability of Critical Points", "Phase Portrait Analysis" ] } ## Represent the system in matrix form $$ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} -\mu & 1 \ -1 & \mu \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} $$ ## Find the trace and determinant of the coefficient matrix $$ \begin{aligned} \tau &= \text{tr}(A) = -\mu + \mu = 0 \ \Delta &= \det(A) = (-\mu)(\mu) - (1)(-1) = -\mu^2 + 1 \end{aligned} $$ ## Apply conditions for a center $$ \begin{aligned} &\tau = 0 \quad \text{(always satisfied)} \ &\Delta > 0 \implies 1 - \mu^2 > 0 \implies \mu^2 < 1 \implies |\mu| < 1 \end{aligned} $$ Determine minimal conditions on the real constant $\mu$ so that $(0, 0)$ is a center for the linear system $$ \begin{aligned} x' &= -\mu x + y \ y' &= -x + \mu y. \end{aligned} $$ $|\mu|$ $<$ $1$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Linear Systems of ODEs" ] }

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Question

Explanation:

Represent the system in matrix form

Find the trace and determinant of the coefficient matrix

Apply conditions for a center

Answer: