Labs / Mathematics
Slope Fields
Every little dash shows the slope dy/dx that a solution must have if it passes through that point. Pick an equation, then click anywhere to drop a starting point — the amber curve is a solution stitched together step by step with Euler's method. Drag the point around and watch the whole solution reshape.
dy/dx = x − ystart —y(right) —
What to try
- For dy/dx = y, drop two starting points a hair apart. Why do their curves fly away from each other so fast?
- Switch to dy/dx = −x / y. What shape do the solution curves trace, and what do they have to do with circles?
- Crank the step size all the way up. Where does the traced curve peel away from the field it's supposed to follow, and why?
- On dy/dx = y(1 − y), start above 1, below 1, and right at 1. Which value does every curve seem to drift toward?
- Turn the field density up and down. Does the underlying equation change, or only how densely you're sampling it?