Question

without solving explicitly, classify the critical points of the given first-order autonomous differential equation as either asymptotically stable or unstable. all constants are assumed to be positive. (enter the critical points for each stability category as a comma-separated list. if there are no critical points in a certain category, enter none.)

\\\frac{dx}{dt} = k(\alpha - x)(\beta - x), \quad \alpha > \beta\\

asymptotically stable \\(x =\\)

unstable \\(x =\\)

Explanation:

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<reasoning>

Identify critical points

Using the Asymptotic Behavior of ODEs knowledge point
\[

$$\begin{aligned} &f(x) = k(\alpha - x)(\beta - x) = 0 \\ &x = \alpha, \quad x = \beta \end{aligned}$$

\]

Compute derivative of the rate function

Using the Asymptotic Behavior of ODEs knowledge point
\[

$$\begin{aligned} &f(x) = k(x^2 - (\alpha + \beta)x + \alpha\beta) \\ &f'(x) = k(2x - (\alpha + \beta)) \end{aligned}$$

\]

Evaluate stability at critical points

Using the Asymptotic Behavior of ODEs knowledge point
\[

$$\begin{aligned} &f'(\alpha) = k(2\alpha - \alpha - \beta) = k(\alpha - \beta) > 0 \quad (\text{since } \alpha > \beta, k > 0) \\ &f'(\beta) = k(2\beta - \alpha - \beta) = k(\beta - \alpha) < 0 \quad (\text{since } \alpha > \beta, k > 0) \end{aligned}$$

\]

Classify stability

Using the Asymptotic Behavior of ODEs knowledge point
\[

$$\begin{aligned} &f'(\beta) < 0 \implies x = \beta \text{ is asymptotically stable} \\ &f'(\alpha) > 0 \implies x = \alpha \text{ is unstable} \end{aligned}$$

\]
</reasoning>

<answer>
Without solving explicitly, classify the critical points of the given first-order autonomous differential equation as either asymptotically stable or unstable. All constants are assumed to be positive.

\[\frac{dx}{dt} = k(\alpha - x)(\beta - x), \quad \alpha > \beta\]

asymptotically stable \(x =\) <blank>\(\beta\)</blank>

unstable \(x =\) <blank>\(\alpha\)</blank>
</answer>

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Answer:

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<reasoning>

Identify critical points

Using the Asymptotic Behavior of ODEs knowledge point
\[

$$\begin{aligned} &f(x) = k(\alpha - x)(\beta - x) = 0 \\ &x = \alpha, \quad x = \beta \end{aligned}$$

\]

Compute derivative of the rate function

Using the Asymptotic Behavior of ODEs knowledge point
\[

$$\begin{aligned} &f(x) = k(x^2 - (\alpha + \beta)x + \alpha\beta) \\ &f'(x) = k(2x - (\alpha + \beta)) \end{aligned}$$

\]

Evaluate stability at critical points

Using the Asymptotic Behavior of ODEs knowledge point
\[

$$\begin{aligned} &f'(\alpha) = k(2\alpha - \alpha - \beta) = k(\alpha - \beta) > 0 \quad (\text{since } \alpha > \beta, k > 0) \\ &f'(\beta) = k(2\beta - \alpha - \beta) = k(\beta - \alpha) < 0 \quad (\text{since } \alpha > \beta, k > 0) \end{aligned}$$

\]

Classify stability

Using the Asymptotic Behavior of ODEs knowledge point
\[

$$\begin{aligned} &f'(\beta) < 0 \implies x = \beta \text{ is asymptotically stable} \\ &f'(\alpha) > 0 \implies x = \alpha \text{ is unstable} \end{aligned}$$

\]
</reasoning>

<answer>
Without solving explicitly, classify the critical points of the given first-order autonomous differential equation as either asymptotically stable or unstable. All constants are assumed to be positive.

\[\frac{dx}{dt} = k(\alpha - x)(\beta - x), \quad \alpha > \beta\]

asymptotically stable \(x =\) <blank>\(\beta\)</blank>

unstable \(x =\) <blank>\(\alpha\)</blank>
</answer>

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