Differential equations, seen geometrically

Idea

The equation does not give the value of $y$ directly. It gives a direction of change at every point of the plane, and a solution curve is the path you get by integrating along that direction.

Interaction

• Slope field: draw direction arrows across the window and turn the differential equation into a visible “map of directions”
• Initial conditions: click to add a starting point and watch how different initial values give different paths
• Euler’s method: integrate step by step with an adjustable step size. As with a Riemann sum, the finer the step, the better the approximation

Suggested order: look at the slope field first → change the initial value and compare paths → make the step larger and watch the error of Euler’s method.

What to notice

• A differential equation first gives a direction at each point, not a ready-made $y(x)$. A solution curve is just the path traced along that one slope field.
• A different starting point follows a different curve in the same field. A step that is too coarse uses a polyline to approximate the same field; it does not switch to a different equation.
• The same field can also be read as the streamlines of a vector field. A path tangent to the field and step-by-step integration are the geometric and numerical versions of the same thing.

Related

• Tractrix
• Vector field streamlines
• Phasor diagram
• Equiangular spiral

Further reading

• Slope field (Wikipedia)
• Euler method (Wikipedia)