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Solving ordinary differential equations in terms of series with real exponents

Authors: D. Yu. GrigorЬev and M. F. Singer
Journal: Trans. Amer. Math. Soc. 327 (1991), 329-351
MSC: Primary 12H05; Secondary 12D15
MathSciNet review: 1012519
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Abstract: We generalize the Newton polygon procedure for algebraic equations to generate solutions of polynomial differential equations of the form $ \sum\nolimits_{i = 0}^\infty {{\alpha _i}{x^{{\beta _i}}}} $ where the $ {\alpha _i}$ are complex numbers and the $ {\beta _i}$ are real numbers with $ {\beta _0} > {\beta _1} > \cdots $. Using the differential version of the Newton polygon process, we show that any such a series solution is finitely determined and show how one can enumerate all such solutions of a given polynomial differential equation. We also show that the question of deciding if a system of polynomial differential equations has such a power series solution is undecidable.

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Keywords: Differential equations, series solutions, Newton polygon
Article copyright: © Copyright 1991 American Mathematical Society

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