<p>This study investigates a nonlinear boundary value problem governing the transient flow of gas through a semi-infinite porous medium, initially maintained at a uniform pressure. The system is modeled by a second-order differential equation subject to boundary conditions at the origin and infinity, with an explicit dependence on a real parameter. The analysis rigorously establishes the existence and uniqueness of the solution by employing the classical fixed-point theorem. Furthermore, a highly accurate analytical approximation of the solution is derived, providing significant insights into the dynamical behavior of the system and its dependence on the governing parameters.</p>