A one-dimensional mathematical model is developed for an unsteady incompressible laminar flow in a circular pipe subjected to a rapid change in the flow rate from an initial flow with flow rate, Qi, to a final flow with flow rate, Qf, in a step-like fashion at an arbitrary time, tc. The change in the flow rate may either be an increment, Qf > Qi, or a decrement, Qf < Qi. The change time, tc, may either belong to the initial flow remaining in a temporally developing state or temporally developed state. The developed model is solved using the Laplace transform method to deduce generalized analytical expressions for the flow characteristics, viz., velocity, pressure gradient, wall shear stress, and skin friction factor, CfRe, where Re is Reynolds number based on the cross-sectional area-averaged velocity and pipe radius. Exact solutions for λa=Qi/Qf=0 and λd=Qf/Qi=0 with tc≥tsi are available in the literature and the present generalized analytical solutions fill the remaining range of parameters, 0<λa<1 and 0<λd<1 with 0<tc<tsi and tc≥tsi, where tsi is the time at which the initial flow reaches the temporally developed state. Exact solutions for canonical pipe flow problems reported in the literature are deduced as subsets of the derived generalized solutions. The parametric study reveals the effects of varying λa or λd and tc on the quantities of practical importance, viz., τs and CfRe, where τs is the time required for the final flow to reach the temporally developed state.