Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 14 / b

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- Solution b

Solution

Substitute the partial derivatives obtained above. The first-order terms cancel exactly:

u_{xx} - \frac{1}{2} u_{y} - y u_{yy} = U_{ξξ} + 2 U_{ξ η} + U_{ηη} - \frac{U _{ξ} - U _{η}}{2 y} - (U_{ξξ} - 2 U_{ξ η} + U_{ηη} - \frac{U _{ξ} - U _{η}}{2 y}) = 4 U_{ξ η} .

The transformed partial differential equation is therefore

U_{ξ η} = 0, ξ > η .

In these characteristic coordinates, the mixed partial derivative is the only surviving term.

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