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arXiv math/02043292002-04math.CAmath-phmath.MP

Differential equations of order 1 in fields of zero characteristic

Jerzy Stryla

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

英文摘要：

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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