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In this paper we present a method to solve the Gdwarf problem in the frame-work of analytical models (based on the instantaneous recycling approximation, IRA).We consider a one-zone closed model without inflows or outflows.We suppose a time-dependent Initial Mass Function (IMF) and we find an integral-differential equation which must be satisfied in order to honour the G-dwarf metallicity distribution as a function of the oxygen abundance. IMFs with one and two slopes are given and discussed also in the framework of a numerical chemical evolution model without IRA.We conclude that it is difficult to reproduce other observational constraints besides the G-dwarf distribution (such as [ O Fe] vs [Fe H ]), and that an IMF with two slopes, with time-dependent shape at the low mass end, would be required. However, even in this case the predicted oxygen gradient along the disk is flat and radial flows would be required to reproduce the observed gradient.