Slope field

A grid of short line segments, each showing $dy/dx$ at that point — a visual map of a differential equation's solutions.

A slope field (direction field) plots a short line segment at many points $(x,y)$ in the plane, each with slope equal to $\frac{dy}{dx}$ as given by the differential equation at that point. Without solving the equation at all, tracing a smooth curve that stays tangent to the segments at every point sketches an approximate solution — the field reveals the SHAPE of every solution simultaneously.

Examples

For $y'=x-y$ , the segment at $(2,2)$ has slope $2-2=0$ (horizontal), while at $(3,0)$ it has slope $3-0=3$ (steep).