Sketching Solutions Through

Differential Equations · Axiom Academy

EXAMPLE Sketching a Solution Through a Point Given a direction field, trace the solution curve through a specific initial point. For the differential equation , sketch the solution curve that passes through the point (1, 2) . The blue segments show the slope at each grid point; the orange curve is the solution through (1,2) , traced in both directions. Nice work — you traced a solution curve through a specific point using only the direction field, with no analytic solving required. An initial condition picks one curve: among the infinitely many solutions to a differential equation, a point like (1,2) singles out exactly one. The slope comes from the equation: substitute the point's coordinates into dy/dx to get the curve's direction right there. Follow the field like a path: starting at the point, move through the segments in both directions to build the sketch. The picture can mislead if you stop early: here the curve briefly flattens near but does not level off — it keeps rising, approaching a slope of +1 . Direction fields give geometric insight before you ever solve a differential equation analytically. Practice tracing curves through other initial points to sharpen the skill.

This is the written version of the interactive lesson above. See the full Differential Equations course.