1. What is Phase Line Analysis?

Phase line analysis is a graphical method for understanding the behavior of one-dimensional autonomous differential equations of the form dx/dt = f(x). It helps identify equilibrium points and determine their stability.

2. How Does the Calculator Work?

The calculator analyzes the differential equation:

Where:

• \( x \) — The variable of interest
• \( f(x) \) — A function of x only (autonomous system)

Explanation: The calculator finds critical points where f(x) = 0, analyzes the sign of f(x) between these points, and determines the stability of each equilibrium.

3. Importance of Phase Line Analysis

Details: Phase line analysis provides a quick visualization of the long-term behavior of solutions without solving the differential equation explicitly. It's particularly useful for understanding population models, chemical reactions, and other autonomous systems.

4. Using the Calculator

Tips: Enter a function f(x) using standard mathematical notation (e.g., "x*(1-x)" for the logistic equation). Specify the range of x values to analyze. The calculator will identify critical points and their stability.

5. Frequently Asked Questions (FAQ)

Q1: What types of functions can I analyze?
A: The calculator works with polynomial, exponential, trigonometric, and logarithmic functions. For best results, use continuous functions.

Q2: How are critical points determined?
A: The calculator numerically finds points where f(x) = 0 within the specified range.

Q3: How is stability determined?
A: Stability is determined by the sign of f(x) near the critical point. If f(x) changes from positive to negative, it's stable. If negative to positive, unstable.

Q4: What if my function has no critical points?
A: The phase line will show uniform behavior (always increasing or always decreasing) across the specified range.

Q5: Can I analyze non-autonomous equations?
A: No, phase line analysis only works for autonomous equations (where f depends only on x, not t).