1. What Is Phase Line Analysis?

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

2. How To Use This Calculator

Simply enter:

• The function f(x) in terms of x (e.g., "x*(1-x)" for logistic growth)
• The range of x values you want to analyze

The calculator will identify equilibrium points and classify them as stable or unstable.

3. Understanding The Results

Equilibrium Points: Values of x where f(x) = 0.
Stable Equilibrium: Arrows on phase line point toward it.
Unstable Equilibrium: Arrows on phase line point away from it.

4. Examples

Example 1: Logistic Growth
dx/dt = x*(1-x)
Equilibrium at x=0 (unstable) and x=1 (stable)

Example 2: Simple Decay
dx/dt = -x
Only equilibrium at x=0 (stable)

5. Frequently Asked Questions (FAQ)

Q1: What types of equations can this analyze?
A: First-order autonomous equations of the form dx/dt = f(x).

Q2: Can I use trigonometric functions?
A: Yes, functions like sin(x), cos(x) are supported.

Q3: How accurate are the equilibrium points?
A: They are calculated numerically with precision to 4 decimal places.

Q4: What if my function has many equilibrium points?
A: The calculator will identify all equilibrium points in your specified range.

Q5: Can I save my results?
A: Currently you need to copy/paste the results manually.