Slope Field Generator

A first-order differential equation dy/dx = f(x, y) tells you the slope of a solution at every point of the plane, even when you cannot write the solution down. A slope field draws that information directly: on a grid of points, each short segment is tilted to slope f(x, y). Because every solution curve must be tangent to the field wherever it passes, the picture shows the whole family of solutions at once, and a particular solution is the curve you get by following the segments from an initial condition.

This tool turns your expression into a compiled function and evaluates it at every grid point to draw the field. When you click, it integrates the equation numerically from that point. Rather than stepping in x, which fails where the slope is vertical, it steps along the curve’s arc length using the direction (1, f)/√(1 + f²) and a fourth-order Runge–Kutta scheme, marching both forward and backward until the curve leaves the window or the equation becomes undefined. For smooth equations the result matches the exact solution to several decimal places inside the window.

A phase portrait is the two-dimensional analogue for autonomous systems dx/dt = f(x, y), dy/dt = g(x, y), such as a damped pendulum, a predator–prey model, or a linear system written in matrix form. Here the arrows show the velocity vector at each point and trajectories are curves parameterised by time. Their qualitative shape is organised by the equilibrium points, where f and g are both zero. The tool seeds Newton’s method from a coarse grid to locate them and then linearises: with the Jacobian’s trace τ and determinant Δ, a negative Δ is a saddle, positive Δ with τ² ≥ 4Δ is a node, and positive Δ with complex eigenvalues is a spiral, stable when τ < 0 and unstable when τ > 0. A zero trace with complex eigenvalues is a center of the linearisation, which nonlinear terms may turn into a very slow spiral.

Reading the pictures is most of the skill. In a slope field, horizontal segments mark where y′ = 0, the isocline along which solutions have maxima, minima, or plateaus; the logistic equation y′ = y(1 − y) shows solutions flattening toward the equilibrium y = 1 and away from y = 0. In a phase portrait, closed orbits mean periodic behaviour, trajectories converging on a point mean a stable equilibrium, and the stable and unstable curves through a saddle divide the plane into regions with different long-term fates.

Numerical curves are approximations. Near a singularity, for example y = 0 in y′ = −x/y, the integration stops rather than inventing values, and very stiff systems can drift over long trajectories. Narrowing the window or comparing against a known solution is the usual check. The plots are meant to accompany the analytic work in a course, not replace it.

Built and maintained by the runcell.dev team. Last reviewed September 2026.