On first integrals for polynomial differential equations on the line

Henryk Żołądek

Studia Mathematica (1993)

  • Volume: 107, Issue: 2, page 205-211
  • ISSN: 0039-3223

Abstract

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We show that any equation dy/dx = P(x,y) with P a polynomial has a global (on ℝ²) smooth first integral nonconstant on any open domain. We also present an example of an equation without an analytic primitive first integral.

How to cite

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Żołądek, Henryk. "On first integrals for polynomial differential equations on the line." Studia Mathematica 107.2 (1993): 205-211. <http://eudml.org/doc/216029>.

@article{Żołądek1993,
abstract = {We show that any equation dy/dx = P(x,y) with P a polynomial has a global (on ℝ²) smooth first integral nonconstant on any open domain. We also present an example of an equation without an analytic primitive first integral.},
author = {Żołądek, Henryk},
journal = {Studia Mathematica},
keywords = {polynomial differential equations; first integral},
language = {eng},
number = {2},
pages = {205-211},
title = {On first integrals for polynomial differential equations on the line},
url = {http://eudml.org/doc/216029},
volume = {107},
year = {1993},
}

TY - JOUR
AU - Żołądek, Henryk
TI - On first integrals for polynomial differential equations on the line
JO - Studia Mathematica
PY - 1993
VL - 107
IS - 2
SP - 205
EP - 211
AB - We show that any equation dy/dx = P(x,y) with P a polynomial has a global (on ℝ²) smooth first integral nonconstant on any open domain. We also present an example of an equation without an analytic primitive first integral.
LA - eng
KW - polynomial differential equations; first integral
UR - http://eudml.org/doc/216029
ER -

References

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  1. [1] V. I. Arnold and S. Yu. Ilyashenko, Ordinary Differential Equations, Dynamical Systems I, Encyclopedia Math. Sci. 1, Springer, Berlin, 1988. (Russian original: Fundamental Directions, Dynamical Systems I, VINITI, Moscow, 1985.) 
  2. [2] I. Bendixson, Sur les courbes définies par des équations différentielles, Acta Math. 25 (1901), 1-88. 
  3. [3] W. Kaplan, Analytic first integrals of ordinary differential equations, Comment. Math. Helv. 47 (1972), 205-212. Zbl0247.34008
  4. [4] W. Kaplan, On partial differential equations of first order, in: Topics in Analysis, Jyväskylä 1970, O. Lehto et al. (eds.), Lecture Notes in Math. 419, Springer, Berlin, 1974, 221-231. 
  5. [5] J.-F. Mattei et R. M. Moussu, Holonomie et intégrales premières, Ann. Sci. Ecole Norm. Sup. (4) 13 (1980), 469-513. Zbl0458.32005
  6. [6] J. Szarski, Sur un problème de caractère intégral relatif à l’équation : ∂ z / ∂ x + Q ( x , y ) ∂ z / ∂ y = 0 définie dans le plan tout entier, Ann. Soc. Polon. Math. 19 (1946), 106-132. Zbl0031.30803
  7. [7] Z. Szmydtówna, Sur les intégrales premières de l'équation y' = f(x,y), ibid. 23 (1950), 167-182. Zbl0041.05403
  8. [8] T. Ważewski, Sur un problème de caractère intégral relatif à l’équation ∂ z / ∂ x + Q ( x , y ) ∂ z / ∂ y = 0 , Mathematica (Cluj) 8 (1933), 103-116. Zbl0008.39403

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