Solving ordinary differential equations in terms of series with real exponents

Abstract

We generalize the Newton polygon procedure for algebraic equations to generate solutions of polynomial differential equations of the form ∑ i = 0 ∞ α i x β i \sum \nolimits _{i = 0}^\infty {{\alpha _i}{x^{{\beta _i}}}} where the α i {\alpha _i} are complex numbers and the β i {\beta _i} are real numbers with β 0 > β 1 > ⋯ {\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.