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Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 31C / Solution Created 2026-09-24 Updated 2026-09-24

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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 33D / b / Solution 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} .

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