Concept preview

What you will be able to do

1. 01Trace a particular solution through a given initial point by following the slope-field segments, keeping the curve tangent to the field and never cutting across a segment.
2. 02Read a field's long-term behaviour: locate equilibrium solutions where $\dfrac{dy}{dx}=0$, and predict where a traced solution levels off, grows without bound, or decays.
3. 03Determine whether a solution is increasing or decreasing from the sign of $\dfrac{dy}{dx}$, and infer concavity by asking how the slope changes along the curve.

To trace the solution through a given point, put your pencil on the point and let the field steer. The segment at your location gives the slope, so you move a short way in that direction — up if the segment tips up, down if it tips down, steeply if it is steep. When you reach the next grid point you read its segment, which may point a little differently, and you adjust your heading to match. The curve you draw is smooth and always tangent to the local segments; it threads between them like a boat following a current. The one absolute rule is that the curve must stay tangent to the field — it may never cut across a segment at an angle, because at every point the solution has exactly the slope the field prescribes there.