Enter any ODE system and explore its behavior
This is your sandbox for exploring differential equations. Type in any ODE system and watch the phase plane come alive with vector fields, nullclines, and trajectories. Use the presets to jump into classic systems, or experiment with your own expressions.
Expressions support standard math notation: variables x, y, operators + - * / ^, functions sin cos exp sqrt abs log, and constants pi e. Click on the canvas to add trajectories from any initial condition.
Enter a two-dimensional autonomous system dx/dt = f(x, y) and dy/dt = g(x, y). The vector field shows the flow direction at each point. Toggle nullclines to see where each component of the velocity is zero -- equilibria occur at their intersections. Click to launch trajectories and observe the long-term behavior.
Click anywhere on the phase plane to place an initial condition and watch the trajectory evolve via RK4 integration.
Enter a single first-order ODE dy/dx = f(x, y) and explore its slope field. The line segments indicate the slope of the solution through each point. Click to place initial conditions and trace solution curves forward and backward using fourth-order Runge-Kutta. Adjust the density to see more or fewer field lines.
Click on the slope field to place initial conditions. Solution curves are traced using RK4 in both forward and backward directions.