Slope Fields

Calculus 2 · Axiom Academy

See every solution of a differential equation at once — without ever solving it. The equation never moves — it just reports a number at each point. For , the reported slope at a point is simply its x -coordinate. Walk a point across the plane and, at each grid location, stamp a short segment tilted to that slope. Stamp enough of them and the "flow" of solutions appears. The slope a solution would have, point by point Because the slope here depends only on x , every segment in a vertical column is parallel. Reading left to right, the segments swing from steeply down , flatten to horizontal at x=0 , then tilt steeply up — and that is exactly the shape of a parabola. Slope = x < 0 : segments tilt downward. At x=-1 the tilt is exactly -1 (a drop). Slope = 0 : segments lie flat. Every solution curve is momentarily level on the y -axis. Slope = x > 0 : segments tilt upward. At x=1 the tilt is exactly +1 (a rise). Integrating gives — a parabola tangent to the field at every point. The constant C slides the curve Different choices of C stack the same parabola at different heights. Each one is a solution; the field they share is identical, because the slope x never depended on y . 3. When the Slope Depends on y Too

This is the written version of the interactive lesson above. See the full Calculus 2 course.