 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-115/2/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

\overset{x}{˙} = α, \overset{y}{˙} = β, \overset{z}{˙} = x β .

For an arbitrary starting point

(x_{0}, y_{0}, z_{0})

x (t) = x_{0} + \int_{0}^{t} α (s) d s, y (t) = y_{0} + \int_{0}^{t} β (s) d s, z (t) = z_{0} + \int_{0}^{t} (x_{0} + \int_{0}^{s} α (r) d r) β (s) d s .

This is the control equation for the Heisenberg horizontal distribution. The final coordinate records a signed area-like integral, so changing the path in the

(x, y)

plane can change the endpoint in the central direction.

 New to topics? Read the docs here!