Sketching Slope Fields

A differential equation like dy/dx = x - y gives you a rule for the slope at any point (x, y), before you know which solution curve passes through it. A slope field turns that rule into a picture: pick a grid of points, plug each one into the right-hand side, and draw a short segment with that slope centered on the point. Do this at enough points and the tiny dashes line up into flowing patterns. Any solution curve must be tangent to the segments it passes through, so the field shows the shape of every possible solution at once, without ever solving the equation.

To compute a single segment, take the point's coordinates and substitute them into dy/dx. For dy/dx = x - y, the point (1, 3) gives slope 1 - 3 = -2, so you draw a short segment through (1,3) tilted at slope -2. Repeat across a grid and you start to see curves emerge: steep in some regions, flat where the right-hand side is zero, mirrored where it changes sign. A common trap is drawing the one curve that looks like it belongs there and mistaking the whole field for that solution. The field has no single starting point built in; it represents infinitely many solutions, each one picked out only once you add an initial condition.

Slope fields set up the rest of the unit. Once you can read a field, separation of variables gives you the actual equation for the curve through a chosen point, and Euler's method uses the same slope-at-a-point idea to step numerically along one specific solution.