 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/a/iv/solution

Differentiate

u_{y} = y^{- 1/2} (U_{ξ} - U_{η})

, retaining the derivative of the prefactor. The product rule and multivariable chain rule give

u_{yy} = - \frac{1}{2 y ^{3/2}} (U_{ξ} - U_{η}) + \frac{1}{y} (U_{ξξ} - 2 U_{ξ η} + U_{ηη}) .

Using

y = (ξ - η)^{2} /16

, this becomes

u_{yy} = \frac{16}{( ξ - η ) ^{2}} (U_{ξξ} - 2 U_{ξ η} + U_{ηη}) - \frac{32}{( ξ - η ) ^{3}} (U_{ξ} - U_{η}) .

The last term is essential because the coordinate change is nonlinear.

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