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:

Identify critical points

Using the Asymptotic Behavior of ODEs knowledge point

$$
LATEXBLOCK0
$$

Compute derivative of the rate function

Using the Asymptotic Behavior of ODEs knowledge point

$$
LATEXBLOCK1
$$

Evaluate stability at critical points

Using the Asymptotic Behavior of ODEs knowledge point

$$
LATEXBLOCK2
$$

Classify stability

Using the Asymptotic Behavior of ODEs knowledge point

$$
LATEXBLOCK3
$$

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>