An adaptation of the Newton Method is used to solve the combined system of ordinary differential and algebraic equations that describe the spatially varied flows that occur from lateral inflow with periodic outflow along open channels. The application selected is lateral inflow into gutters with periodically spaced grates in its bottom through which the accumulated flow is discharged. The occurrence of critical flow may allow the lateral inflow to be solved as a separate problem, and the outflow to be handled separately. Where subcritical flow exists, the composite system of differential and algebraic equations resulting from a series of gutter-grates must be solved simultaneously. Solutions to such example applications are given.