1. What is Phase Line Analysis?

Phase line analysis is a graphical method for studying autonomous differential equations of the form dy/dt = f(y). It helps identify equilibrium points and their stability without solving the differential equation explicitly.

2. How Does the Calculator Work?

The calculator analyzes the differential equation:

Where:

• \( y \) — Dependent variable
• \( t \) — Independent variable (often time)
• \( f(y) \) — Function of y only (autonomous equation)

Explanation: The calculator finds roots of f(y) (equilibrium points) and analyzes the sign of f(y) between these points to determine stability.

3. Importance of Phase Line Analysis

Details: Phase line analysis provides quick insights into the long-term behavior of solutions without requiring exact solutions. It's particularly useful for nonlinear equations where analytical solutions may be difficult or impossible to find.

4. Using the Calculator

Tips: Enter a function of y only (autonomous equation), specify the range of y values to analyze. The calculator will identify equilibrium points and their stability characteristics.

5. Frequently Asked Questions (FAQ)

Q1: What types of equations can be analyzed?
A: Only autonomous equations (where dy/dt depends only on y, not on t directly).

Q2: How are equilibrium points determined?
A: Equilibrium points are values of y where f(y) = 0 (where dy/dt = 0).

Q3: How is stability determined?
A: If f(y) > 0, solutions move upward; if f(y) < 0, solutions move downward. Equilibrium is stable if solutions approach it, unstable if they move away.

Q4: What are the limitations of phase line analysis?
A: It only shows qualitative behavior, not exact solutions. It doesn't work for non-autonomous equations or systems of equations.

Q5: Can I see the actual phase line diagram?
A: In a full implementation, this would include a graphical phase line showing equilibrium points and flow directions.