This paper analyses the classical problem of transient surface-groundwater interaction in a stream-aquifer system under rather realistic conditions. The downward sloping unconfined aquifer is in contact with a constant water level at one end, and a fully penetrating stream at the other end whose water level is rising at a uniform speed. Furthermore, the aquifer is replenished by a vertical time-varying recharge. Closed form analytical expressions for hydraulic head and flow rate in the aquifer are obtained by solving the linearized Boussinesq equation using Laplace transform method. Effects of aquifer parameters on transient water table and flow rate are illustrated with a numerical example. To assess the efficiency of the linearization method, analytical solutions are compared with numerical solutions of the corresponding non-linear equation.