Question

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|\\) ?

Explanation:

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<reasoning>

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

\[

$$\begin{aligned} &\tau = 0 \quad \text{(always satisfied)} \\ &\Delta > 0 \implies 1 - \mu^2 > 0 \implies \mu^2 < 1 \implies |\mu| < 1 \end{aligned}$$

\]
</reasoning>

<answer>
Determine minimal conditions on the real constant \(\mu\) so that \((0, 0)\) is a center for the linear system
\[

$$\begin{aligned} x' &= -\mu x + y \\ y' &= -x + \mu y. \end{aligned}$$

\]
\(|\mu|\) <blank>\(<\)</blank> <blank>\(1\)</blank>
</answer>

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"question_type": "Fill-in-the-blank",
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"Mathematics",
"Calculus",
"Linear Systems of ODEs"
]
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</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

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

\[

$$\begin{aligned} &\tau = 0 \quad \text{(always satisfied)} \\ &\Delta > 0 \implies 1 - \mu^2 > 0 \implies \mu^2 < 1 \implies |\mu| < 1 \end{aligned}$$

\]
</reasoning>

<answer>
Determine minimal conditions on the real constant \(\mu\) so that \((0, 0)\) is a center for the linear system
\[

$$\begin{aligned} x' &= -\mu x + y \\ y' &= -x + \mu y. \end{aligned}$$

\]
\(|\mu|\) <blank>\(<\)</blank> <blank>\(1\)</blank>
</answer>

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"Mathematics",
"Calculus",
"Linear Systems of ODEs"
]
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