  We describe a solving semi-decision method based on examination of the
rational structures of the generalized integrating factors of first-order ODEs.
We propose a conjecture that for some family of equations of the type
dy/dx=P(x,y)/Q(x,y), with P and Q polynomials only in y (or in x), the general
form of the structures of generalized integrating factors are rational in y (or
in x). In such a way one can obtain a differential-algebraic polynomial system
for undetermined parameters of the structures. The successful solution of this
system (it is sufficient to find any particular solution) automatically leads
to finding the general solutions of ODEs.
