1. What is a Phase Line Diagram?

A phase line diagram is a graphical tool used to analyze one-dimensional autonomous differential equations of the form dy/dt = f(y). It shows the equilibrium points and their stability without solving the differential equation.

2. How Does the Calculator Work?

The calculator analyzes the sign of f(y) to determine:

Key Features:

• Identifies all equilibrium points in the specified range
• Classifies each equilibrium as stable or unstable
• Shows the direction field between equilibrium points

3. Importance of Phase Line Analysis

Details: Phase line diagrams provide quick qualitative understanding of long-term behavior of solutions without solving the differential equation. They are particularly useful for population models, chemical reactions, and other autonomous systems.

4. Using the Calculator

Tips:

• Enter a function of y (e.g., "y*(1-y)" for logistic growth)
• Specify a reasonable range for y values
• The calculator will display equilibrium points and their stability

5. Frequently Asked Questions (FAQ)

Q1: What types of functions can I analyze?
A: Any one-dimensional autonomous ODE where dy/dt depends only on y (not on t).

Q2: How are equilibrium points determined?
A: Values of y where f(y) = 0 are equilibrium points.

Q3: How is stability determined?
A: If f(y) changes from positive to negative at the equilibrium, it's stable. If negative to positive, it's unstable.

Q4: Can I analyze non-autonomous equations?
A: No, phase line analysis only works for autonomous equations (no explicit t-dependence).

Q5: What if my function has parameters?
A: You can include parameters (e.g., "r*y*(1-y/K)"), but you'll need to specify their values.