1. What is Phase Line Analysis?

Phase line analysis is a graphical method for understanding the behavior of solutions to autonomous differential equations of the form dy/dt = f(y). It helps identify equilibrium points and their stability.

2. How Does the Calculator Work?

The calculator analyzes the differential equation:

Where:

• \( y \) — Dependent variable
• \( t \) — Independent variable (often time)
• \( f(y) \) — Function defining the differential equation

Explanation: The calculator finds equilibrium points (where f(y) = 0) and analyzes their stability by examining the sign of f(y) around these points.

3. Importance of Phase Line Analysis

Details: Phase line analysis provides qualitative understanding of solution behavior without solving the equation explicitly. It's crucial for modeling biological, chemical, and physical systems.

4. Using the Calculator

Tips: Enter the right-hand side of dy/dt = f(y). Specify a range for y values to search for equilibrium points. Use standard mathematical notation (e.g., y^2 for y², exp(y) for eʸ).

5. Frequently Asked Questions (FAQ)

Q1: What are equilibrium points?
A: Values of y where dy/dt = 0. These represent constant solutions to the differential equation.

Q2: How is stability determined?
A: If f(y) changes from positive to negative at an equilibrium, it's stable. If negative to positive, it's unstable.

Q3: What types of equations can be analyzed?
A: Only autonomous equations (where f depends only on y, not t). Non-autonomous equations require different methods.

Q4: Can I analyze systems of equations?
A: This calculator is for single equations. Systems require phase plane analysis.

Q5: What if my equation has parameters?
A: You'll need to enter specific numerical values for all parameters to use this calculator.