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A new look at the local solvability condition of inhomogeneous ordinary differential equations. (English) Zbl 1139.34062

From the introduction: Let \(P = P(z,d/dz)\) be a linear ordinary difierential operator with holomorphic coefficients defined on a neighbourhood \(X\) of the origin \(O\) in \(C\).
We consider the local solvability condition to the inhomogeneous equations \(P(z, d/dz)u(z) = f(z)\) in the space \(\widehat{\mathcal O}_{X,O}\) of formal power series at \(O\).
We propose an alternative approach with an Intention to establish a new effective method to determine the local solvability conditions. For this purpose, we adopt a duality method used by H. Komatsuj [J. Fac. Sci., Univ. Tokyo, Sect. I A 18, 379–398 (1971; Zbl 0232.34026)] in the study of index theorems and hyperfunction solutions of an ordinary differential equation. We develop a complex variable version of the duality method to show that the local solvability condition can be written in terms of residues. Then upon using the concept of local cohomology and the theory of \(D\)-modules of one variable, we derive, in a constructive manner, a regular singular holonornic system of ordinary differential equations supported at the origin that completely describes the local solvability condition.

MSC:

34M25 Formal solutions and transform techniques for ordinary differential equations in the complex domain
12H05 Differential algebra
34A05 Explicit solutions, first integrals of ordinary differential equations

Keywords:

local solutions

Software:

OpenXM
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Full Text: DOI

References:

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