Sketch Phase Portrait Calculator

System of Differential Equations:

\[ \frac{dx}{dt} = f(x,y), \quad \frac{dy}{dt} = g(x,y) \]

dx/dt = f(x,y):

dy/dt = g(x,y):

x Range:

y Range:

Phase Portrait:

Enter the differential equations and ranges above to generate the phase portrait.

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1. What is a Phase Portrait?

A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. It shows how the system evolves over time from various initial conditions, revealing stability properties and behavior patterns.

2. How Does the Calculator Work?

The calculator uses numerical methods to solve the system of differential equations:

\[ \frac{dx}{dt} = f(x,y), \quad \frac{dy}{dt} = g(x,y) \]

Where:

\( x \) — State variable (unitless)
\( y \) — State variable (unitless)
\( t \) — Time (s)
\( f(x,y) \) — Function defining dx/dt
\( g(x,y) \) — Function defining dy/dt

Explanation: The calculator numerically integrates the equations to generate vector fields and trajectories that form the phase portrait.

3. Importance of Phase Portraits

Details: Phase portraits are essential for understanding the qualitative behavior of dynamical systems, identifying equilibrium points, stability analysis, and predicting long-term system behavior without solving equations explicitly.

4. Using the Calculator

Tips: Enter the differential equations using standard mathematical notation. Define appropriate x and y ranges to capture the interesting behavior of the system. Use variables 'x' and 'y' in your equations.

5. Frequently Asked Questions (FAQ)

Q1: What types of equations can I analyze?
A: This calculator can handle autonomous systems of two first-order ordinary differential equations.

Q2: How accurate are the generated phase portraits?
A: The accuracy depends on the numerical method used and the step size. For most educational purposes, the results are sufficiently accurate.

Q3: Can I analyze higher-dimensional systems?
A: This calculator is designed for 2D systems. Higher-dimensional systems require specialized tools and different visualization techniques.

Q4: What should I do if the phase portrait looks incorrect?
A: Check your equations for syntax errors, ensure you're using appropriate ranges, and verify that your equations represent the intended system.

Q5: Can I save or export the phase portrait?
A: This implementation shows the concept. A full implementation would include export options for the generated visualization.