 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-332/1/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

q_{x} = R

. The zero-flux condition at the divide gives

q = R (x - L)

. Substituting the Darcy flux and integrating from the absorbing river boundary yields

u_{b} (\frac{h ^{2}}{2} + \frac{β h ^{3}}{6}) = R (Lx - \frac{x ^{2}}{2}) .

Thus the nonnegative groundwater height is the unique nonnegative root of

h^{2} + \frac{β}{3} h^{3} = \frac{R}{u _{b}} (2 Lx - x^{2}) .

The height increases toward the divide, where

h_{x} = 0

. The river receives all the rainfall, so its steady volume flux per unit width is

Q = R L

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