91Ó°ÊÓ

In differential equations, equilibrium values are the points where the derivative of the function equals zero. Essentially, these are the points where the system does not change. To identify these values, you need to set the given derivative equation to zero. For the equation \( \frac{dy}{dx} = y^2 - 2y \), we set it to zero: \( y(y-2) = 0 \). This factoring yields two possible solutions for \( y \), which are \( y = 0 \) and \( y = 2 \).
These values mean that if a system reaches these points, it will remain there unless disturbed by external factors. In this problem, these equilibrium values help us to analyze the stability and behavior of the system.

Stability analysis helps to determine whether an equilibrium value is stable or unstable. This analysis involves checking the sign of the derivative \( \frac{dy}{dx} \) in the intervals around each equilibrium point. For \( \frac{dy}{dx} = y^2 - 2y \), checking the intervals:

• For \( y < 0 \), \( \frac{dy}{dx} > 0 \), indicating that values are increasing as \( y \) approaches 0. This suggests that \( y = 0 \) is unstable, as solutions tend to move away from this equilibrium.
• For \( 0 < y < 2 \), \( \frac{dy}{dx} < 0 \), which means solutions are decreasing, moving towards \( y = 2 \). This marks \( y = 2 \) as stable, since solutions settle around this point.
• For \( y > 2 \), \( \frac{dy}{dx} > 0 \), demonstrating that values are increasing again, keeping \( y = 2 \) attractive from above.

This analysis reveals the behavior of the system around each equilibrium point, helping to predict the system's long-term behavior.

Phase lines visually represent the behavior of solutions to differential equations. Constructing a phase line is a method to combine equilibrium points and their stability into a single diagram. For our function \( \frac{dy}{dx} = y^2 - 2y \), we previously identified equilibrium points and their stability. The creation of a phase line follows:
Place the equilibrium values, \( y = 0 \) and \( y = 2 \), on a number line. Below and above these points, derive the direction of the solutions from the stability analysis:

• Arrows pointing right (\( \rightarrow \)) from \((-\infty, 0)\) and \((2, \infty)\) indicate increasing solutions.
• Arrows pointing left (\( \leftarrow \)) between \((0, 2)\) show decreasing tendencies.

This visual system helps in understanding how solutions progress and converge towards or diverge from equilibrium points over time, making these behaviors clearer and intuitive.