# Reduction of differential equations

• Volume: 53, Issue: 1, page 199-204
• ISSN: 0137-6934

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## Abstract

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Let (F,D) be a differential field with the subfield of constants C (c ∈ C iff Dc=0). We consider linear differential equations (1) $Ly={D}^{n}y+{a}_{n-1}{D}^{n-1}y+...+{a}_{0}y=0$, where ${a}_{0},...,{a}_{n}\in F$, and the solution y is in F or in some extension E of F (E ⊇ F). There always exists a (minimal, unique) extension E of F, where Ly=0 has a full system ${y}_{1},...,{y}_{n}$ of linearly independent (over C) solutions; it is called the Picard-Vessiot extension of F E = PV(F,Ly=0). The Galois group G(E|F) of an extension field E ⊇ F consists of all differential automorphisms of E leaving the elements of F fixed. If E = PV(F,Ly=0) is a Picard-Vessiot extension, then the elements g ∈ G(E|F) are n × n matrices, n= ord L, with entries from C, the field of constants. Is it possible to solve an equation (1) by means of linear differential equations of lower order ≤ n-1? We answer this question by giving neccessary and sufficient conditions concerning the Galois group G(E|F) and its Lie algebra A(E|F).

## How to cite

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Skórnik, Krystyna, and Wloka, Joseph. "Reduction of differential equations." Banach Center Publications 53.1 (2000): 199-204. <http://eudml.org/doc/209074>.

@article{Skórnik2000,
abstract = {Let (F,D) be a differential field with the subfield of constants C (c ∈ C iff Dc=0). We consider linear differential equations (1) $Ly = D^\{n\}y + a_\{n-1\}D^\{n-1\}y+...+ a_\{0\}y = 0$, where $a_0,... ,a_n ∈ F$, and the solution y is in F or in some extension E of F (E ⊇ F). There always exists a (minimal, unique) extension E of F, where Ly=0 has a full system $y_1,... ,y_n$ of linearly independent (over C) solutions; it is called the Picard-Vessiot extension of F E = PV(F,Ly=0). The Galois group G(E|F) of an extension field E ⊇ F consists of all differential automorphisms of E leaving the elements of F fixed. If E = PV(F,Ly=0) is a Picard-Vessiot extension, then the elements g ∈ G(E|F) are n × n matrices, n= ord L, with entries from C, the field of constants. Is it possible to solve an equation (1) by means of linear differential equations of lower order ≤ n-1? We answer this question by giving neccessary and sufficient conditions concerning the Galois group G(E|F) and its Lie algebra A(E|F).},
author = {Skórnik, Krystyna, Wloka, Joseph},
journal = {Banach Center Publications},
keywords = {differential algebra; linear differential equations; operational calculus},
language = {eng},
number = {1},
pages = {199-204},
title = {Reduction of differential equations},
url = {http://eudml.org/doc/209074},
volume = {53},
year = {2000},
}

TY - JOUR
AU - Skórnik, Krystyna
AU - Wloka, Joseph
TI - Reduction of differential equations
JO - Banach Center Publications
PY - 2000
VL - 53
IS - 1
SP - 199
EP - 204
AB - Let (F,D) be a differential field with the subfield of constants C (c ∈ C iff Dc=0). We consider linear differential equations (1) $Ly = D^{n}y + a_{n-1}D^{n-1}y+...+ a_{0}y = 0$, where $a_0,... ,a_n ∈ F$, and the solution y is in F or in some extension E of F (E ⊇ F). There always exists a (minimal, unique) extension E of F, where Ly=0 has a full system $y_1,... ,y_n$ of linearly independent (over C) solutions; it is called the Picard-Vessiot extension of F E = PV(F,Ly=0). The Galois group G(E|F) of an extension field E ⊇ F consists of all differential automorphisms of E leaving the elements of F fixed. If E = PV(F,Ly=0) is a Picard-Vessiot extension, then the elements g ∈ G(E|F) are n × n matrices, n= ord L, with entries from C, the field of constants. Is it possible to solve an equation (1) by means of linear differential equations of lower order ≤ n-1? We answer this question by giving neccessary and sufficient conditions concerning the Galois group G(E|F) and its Lie algebra A(E|F).
LA - eng
KW - differential algebra; linear differential equations; operational calculus
UR - http://eudml.org/doc/209074
ER -

## References

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14. K. Skórnik and J. Wloka [2] m-Reduction of ordinary differential equations, Coll. Math. 78 (1998), 195-212. Zbl0927.12002
15. K. Skórnik and J. Wloka [3] Some remarks concerning the m-reduction of differential equations, Integral Transforms and Special Functions 9 (2000), 75-80. Zbl1019.12003
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