Differential equations of order 1 in fields of zero characteristic

Details Differential equations of order 1 in fields of zero characteristic Differential equations of order 1 in fields of zero characteristic

2002-04-28

arxiv.org/abs/math/0204329v3

Mathematics

Classical Analysis and ODEs

21 pages, changes: a new proof of Th. 12

Scientific paper

I begin from a particular field of generalised Puiseux series and investigate a class of nonlinear differential equations in the field. It is appeared that the main part of differential equation determines solvability and positions of resonance i.e. the appearance of a free constant in solutions. Secondly, for any singular differential equation in a differential field of the form $Q(y)\dot y=P(y)$, $P,Q$ polynomials. Then the greatest Picard-Vessiot extension exists. It is shown that the arising set of solutions is obtained from algebraic equations labelled by constants. Sufficient conditions for the PV extension being extended liouvillian are delivered.

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