Calculus
Differential Equations
An equation that gives you the slope instead of the value. Draw that slope everywhere and the plane fills with little segments; drop a point anywhere and the solution threads itself through them. Then take steps too big and watch Euler's method drift off the truth.
the solution through your pointparticles riding the fieldEuler's steps
Drag the start point anywhere on the plane
Slope at the start point—
—
The equation y′ =
tiny, faithfulhuge, wrong
Show
Euler steps—
Euler error at the end—
solution exists until—
What to observe
- Drag the dot: the bright curve stays tangent to every segment it passes. Turn on the family and each curve is the same rule from a different start, which is the + C.
- Let the particles loose on x − y: dots from every height pile onto the line y = x − 1. The solutions forget where they started.
- On y(1 − y) the particles empty out of y = 0 and crowd into y = 1. Both lines are solutions that never move: an unstable and a stable equilibrium.
- Turn on Euler's method with a big h: each hop commits to the slope where it starts, so the red path drifts off the true curve. Shrink h and the gap closes.
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