A macroscopic root water uptake model is considered, described by Richards' equation with a state-dependent sink term. After semi-discretizing the time derivative, we integrate in the space the second-order ODEs system. Endowed with Dirichlet boundary conditions, this problem takes the form of a boundary value problem, that can be classically solved by a shooting method. The problem is faced with Gardner's hydraulic functions and Kirchhoff transformation of water content.