 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-1/33d/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Substituting the displayed matrices into

V_{x} - U_{y} + [U, V]

makes every off-diagonal entry vanish. The two nonzero diagonal entries are opposites, the first being

- u_{x y} + e^{u} - e^{- 2 u} .

Thus compatibility is equivalent to the Tzitzeica equation

u_{x y} = F (u) = e^{u} - e^{- 2 u} .

 New to topics? Read the docs here!