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On the differential invariants of a linear ordinary differential equation

Published online by Cambridge University Press:  14 November 2011

Joseph P. S. Kung  and
Gian-Carlo Rota
Joseph P. S. Kung
Affiliation:
Department of Mathematics, North Texas State University, Denton, Texas 76203, U.S.A.
Gian-Carlo Rota
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.
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Synopsis

Let y(n) + a1y(n−1 +…+ an−1y(1) + an = 0 (*) be a linear ordinary differential equation of order n. A (relative) differential invariant of (*) is a differential polynomial function π(xi) defined on the solution space of (*) satisfying: there is an integer g such that for all invertible linear transformations α of V into itself, π(αxi) = (det α)βπ(xi). We prove in a purely algebraic manner the following two theorems: A. The differential invariants of (*) are generated algebraically by the Wronskian W and the coefficients ala2, …, an of (*). B. Every generic differential relation (i.e. differential relation which holds for every linear ordinary differential equation of order n) among W, a1 …, an can be deduced algebraically from Abel's identity, W′ = −a1W. The second theorem may be considered as an algebraic version of the existence theorem for linear ordinary differential equations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 89 , Issue 1-2 , 1981 , pp. 111 - 123
Copyright
Copyright © Royal Society of Edinburgh 1981

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