Direction fields and integral curves of ODEs
Click — one curve. Brush — hold LMB and drag. Wheel — zoom. Parameters in the equation (e.g. a*x + b*y) create sliders automatically.
Click the field to choose a starting point.
| i | x | y | k₁ | k₂ | k₃ | k₄ | xₙ₊₁ | yₙ₊₁ |
|---|
Quick reference
We consider a first-order ODE solved for the derivative:
\[ \dfrac{dy}{dx} = f(x,y) \]The direction field at each point \((x,y)\) is a short dash whose slope is \(f(x,y)\).
An integral curve is the graph of a solution \(y(x)\) that is tangent to the field at every point. By Picard’s theorem, if \(f\) is continuous and Lipschitz in \(y\), exactly one integral curve passes through each point.
Plot a direction field of a differential equation online
The ODEPLOT calculator visualizes a first-order ODE direction field in the browser. Enter \(y' = f(x,y)\) and the app draws slope marks that show the geometry of the equation.
A click on the grid starts an integral curve through \((x_0, y_0)\). Change parameters, pick presets, zoom, and export a PNG.
What you can do on this page
On this page you can expand the canvas to full screen; choose and compare Euler, Heun (RK2), and RK4 on one click; inspect the step table (\(x\), \(y\), and method coefficients) and download the calculation as CSV; animate equation parameters and a stream of points on the field.