An analytical investigation is performed of Petschek-type reconnection in an incompressible plasma medium with a non-uniform magnetic field distribution, given by B=√(2 z ̄) in complex variable notation. The initial two-dimensional (2D) configuration contains a current sheet, modelled as a tangential discontinuity, which runs along the positive x-axis in the complex plane, and which terminates in a Y-type neutral point at one end (x = 0). In the model, reconnection is initiated through the introduction of a tangential electric field component in a localized part of the current sheet referred to as the diffusion region. Outside the diffusion region, which we approximate as an X-type neutral point, the behaviour of the magnetic field and plasma flow is governed by the equations of ideal magnetohydrodynamics (MHD), which for the purpose of this analysis are transformed into a Lagrangian coordinate system. With the restriction of a small reconnection rate, a perturbation method is applied to solve the MHD equations for the outflow region, containing the reconnected plasma, and the surrounding inflow region. The outflow region is treated as a thin boundary layer, and the usual matching procedure for problems of this type is used. In agreement with the results of previous studies of Petschektype reconnection in non-uniform, asymmetric magnetic field and velocity distributions, the boundary and interior structure of the outflow region is found to consist of a combination of Petschek-type shocks and tangential discontinuities. This is in contrast to the case of an initially uniform magnetic field distribution, analysed originally by Petschek, where the outflow region is entirely bounded by Petschek-type shocks and contains no interior discontinuities. The new features arise because the shocks propagate at the local Alfvén velocity in the medium, whereas plasma entering the outflow region is accelerated to the Alfvén velocity as measured at the reconnection site. As a result of the discrepancy between the plasma outflow speed and the shock propagation speed in the model configuration analysed here, the behaviour of the field and flow is qualitatively and quantitatively different on opposite sides of the X-line. In particular, the outflow region propagating into the direction of increasing field strength (i.e. x increasing) is bounded entirely by Petschek-type shocks, but contains a tangential discontinuity which extends the original current sheet through the leading point (or nose) into the interior of the outflow region. On the other hand, the outflow region propagating into the direction of decreasing field strength (x decreasing, but remaining positive) is bounded on the front by tangential discontinuities, which emerge out of Petschek-type shocks, and merge into the original current sheet at the nose. When the reconnection rate drops to zero, there is an additional reverse type of behaviour which occurs in the trailing part; tangential discontinuities develop to form the tail end boundary of the outflow region in the former case, and in the latter case a tangential discontinuity develops which penetrates through the tail into the outflow region. We also investigate the consequences of a reconnection site which moves along the current sheet, analyse what happens when there are two successive pulses of reconnection, and briefly discuss the situation when the outflow region propagates beyond the edge of the current sheet.