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

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- Solution b

Solution

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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