AHL 5.15 (HL)—Slope fields

A slope field places a small line segment with slope dy/dx=f(x,y) at many points. A solution curve follows the local directions and cannot cross itself where the differential equation gives a unique slope.

Equilibria appear where f(x,y)=0, because the segments are horizontal there. Draw trajectories from an initial condition by following the field, while remembering that the sketch is qualitative unless a scale is stated.

For dy/dx=y−x, the line y=x has zero slope segments. A solution starting above that line initially rises more steeply; the field helps predict the direction before numerical work.

A slope field is not the solution curve and its segments are not vectors showing speed. Read slope, position and equilibrium separately.