# Reduction of differential equations

Krystyna Skórnik; Joseph Wloka

Banach Center Publications (2000)

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

## Access Full Article

top## Abstract

top## How to cite

topSkórnik, 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

top- J. E. Humphreys [1] Linear algebraic groups, Springer, Berlin, 1975.
- J. E. Humphreys [2] Introduction to Lie algebras and representation theory, Springer, Berlin, 1972. Zbl0254.17004
- E. L. Ince [1] Ordinary differential equations, Dover Publ., New York, 1956.
- Y. Kaplansky [1] An introduction to differential algebra, Hermann, Paris, 1976. Zbl0083.03301
- E. R. Kolchin [1] Algebraic matrix groups and Picard-Vessiot theory of homogeneous linear ordinary differential equations, Ann. of Math. 49 (1948), 1-42.
- E. R. Kolchin [2] Differential algebra and algebraic groups, Academic Press, New York, 1973. Zbl0264.12102
- S. Lang [1] Algebra, Reading, Addison-Wesley Publ., 1984.
- A. R. Magid [1] Lectures on differential Galois theory, American Math. Soc., 1994.
- J. Mikusiński [1] Operational calculus, Pergamon Press, New York, 1959.
- M. F. Singer [1] Solving homogeneous linear differential equations in terms of second order linear differential equations, Amer. J. Math. 107 (1985), 663-696. Zbl0564.12022
- M. F. Singer [2] Algebraic relations among solutions of linear differential equations: Fano's theorem, Amer. J. Math. 110 (1988), 115-144. Zbl0651.12016
- M. F. Singer [3] An outline of differential Galois theory, in: Computer algebra and differential equations, E. Tournier (ed.), Academic Press, 1989, 3-57.
- K. Skórnik and J. Wloka [1] Factoring and splitting of s-differential equations in the field of Mikusiński, Integral Transforms and Special Functions 4 (1996), 263-274. Zbl0862.34005
- K. Skórnik and J. Wloka [2] m-Reduction of ordinary differential equations, Coll. Math. 78 (1998), 195-212. Zbl0927.12002
- 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
- C. Tretkoff and M. Tretkoff [1] Solution of the inverse problem of differential Galois theory in the classical case, Amer. J. Math. 101 (1979), 1327-1332. Zbl0423.12021
- J. T. Wloka [1] Über lineare s-Differentialgleichungen in der Operatorenrechnung, Math. Ann. 166 (1966), 233-256.
- A. E. Zalesskij [1] Linear groups, Encycl. of Math. Sciences 37 (Algebra IV), Springer, Berlin, 1993.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.