Phase portraits of second-order ODEs
Click — one trajectory. Brush — hold LMB and drag. Wheel — zoom. Parameters in the equation (e.g. w, b) create sliders automatically.
Presets (examples with parameters)
Here v = y′. The phase portrait is drawn in (y, v) coordinates.
Mathematical notation
Quick reference
A second-order ODE:
\[ y'' = f(x,y,y') \]For autonomous equations, \(v = y'\) yields
\[ \begin{cases} y' = v \\ v' = f(y,v) \end{cases} \]The \((y,v)\) plane is the phase plane. Typical portraits: centers (oscillator), saddles (pendulum), limit cycles (van der Pol).
Plot a second-order phase portrait online
ODEPLOT reduces a second-order ODE to a first-order system and draws the vector field on the \((y, y')\) plane.
Presets include the harmonic oscillator, the pendulum, and van der Pol. Click to launch trajectories and save a PNG.