Non-similar solutions of steady two-dimensional and axi-symmetric compressible laminar boundary layer flows with non-uniform slot injection (or suction) have been obtained, from the starting point of the stream-wise co-ordinate to the exact point of separation. The difficulties arising at the starting point of the stream-wise co-ordinate, and at the edges of the slot and at the point of separation have been overcome by applying the method of quasi-linear implicit finite difference scheme, along with an appropriate selection of finer step size along the stream-wise direction. It is observed that the separation can be delayed by a non-uniform slot suction and also by moving the slot downstream but the effect of non-uniform slot injection is just the opposite. The increase of total enthalpy at the wall causes the separation to occur earlier while cooling delays it. The increase of Mach number shifts the point of separation upstream due to the adverse pressure gradient.