Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 332 / 1 / a / Solution

In the steady state, mass conservation reduces to $q_x=R$. The zero-flux condition at the divide gives $q=R(x-L)$. Substituting the Darcy flux and integrating from the absorbing river boundary yieldsThus the nonnegative groundwater height is the unique nonnegative root ofThe height increases toward the divide, where $h_x=0$. The river receives all the rainfall, so its steady volume flux per unit width is $\boxed{Q=RL}$.