 Solution

ID: 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/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} .

U_{ξ η} = 0, ξ > η .

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

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