On a theorem of Bendixson

Abstract

such that the successive application of the transformations of C,, reduces the initial system to a similar one with lowest-degree terms in X or Y, unity. The application of all the C, leads to the detection of all the TO-curves of the initial system. In my first paper [I], I derived anew Bendixson’s fundamental result, but gave a precise method for the detection of the collections C, (with some collapsed transformations). My analysis rested very strongly upon the Weierstrass preparation theorem plus a special choice of coordinate system.