J. Aust. Math. Soc. 83 (2007), no. 2, pp. 157–180.

Transcendental meromorphic solutions of some algebraic differential equations

Katsuya Ishizaki Nobushige Toda
Department of Mathematics
Nippon Institute of Technology
4-1 Gakuendai Miyashiro
Minamisaitama Saitama 345-8501
Japan
ishi@nit.ac.jp
Center for General Education
Aichi Institute of Technology
Yakusa, Toyota-shi
Aichi-ken 470-0392
Japan
toda3-302@coral.ocn.ne.jp
Received 6 December 2005; revised 16 May 2006
Communicated by P. Fenton
Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science, (C) (1) (No. 16540202)

Abstract

In this paper we treat transcendental meromorphic solutions of some algebraic differential equations. We consider the number of distinct transcendental meromorphic solutions. Algebraic relations between meromorphic solutions and comparisons of the growth of transcendental meromorphic solutions are also discussed.

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2000 Mathematics Subject Classification: primary 34A20; secondary 30D35
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References

  1. S. B. Bank, G. G. Gundersen and I. Laine, ‘Meromorphic solutions of the Riccati differential equation’, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), 369–398. MR658934
  2. S. B. Bank and R. P. Kaufman, ‘On the growth of meromorphic solutions of the differential equation (y')^{m}=r(z,y)’, Acta Math. 144 (1980), 223–248. MR573452
  3. S. Elaydi, An Introduction to Difference Equations (Springer-Verlag, New York, 1996). MR1410259
  4. A. A. Gol'dberg, ‘On single-valued solutions of first-order differential equations’, Ukrain. Mat. Zh. 8 (1956), 254–261. MR85396
  5. F. Gross, ‘On the equation f^{n}+g^{n}=1’, Bull. Amer. Math. Soc. 72 (1966), 86–88. MR185125
  6. W. K. Hayman, Meromorphic Functions (Oxford at the Clarendon Press, 1964). MR164038
  7. Y. Z. He and I. Laine, ‘The Hayman–Miles theorem and the differential equation (y')^{n}=r(z,y)’, Analysis (Munich) 10 (1990), 387–396. MR1085804
  8. H. Herold, Differentialgleichungen im Komplexen (Vandenhoeck & Ruprecht, Gottingen, 1975). MR470293
  9. E. Hille, Ordinary Differential Equations in the Complex Domain (Wiley and Sons, New York–London–Sydney–Toronto, 1976). MR499382
  10. I. Laine, ‘On the behavior of the solution of some first order differential equations’, Ann. Acad. Sci. Fenn. Ser. A I 497 (1971). MR387702
  11. I. Laine, Nevanlinna Theory and Complex Differential Equations (W. Gruyter, Berlin–New York, 1992). MR1207139
  12. R. Nevanlinna, Le théorème de Picard-Borel et la théorie des fonctions méromorphes (Gauthier-Villard, Paris, 1929).
  13. N. Steinmetz, Eigenschaften eindeutiger Lösungen gewöhnlicher Differentialgleichungen im Komplexen (Ph.D. Thesis, Karlsruhe, 1978).
  14. N. Steinmetz, ‘Zur theorie der binomischen differentialgleichungen’, Math. Ann. 5 (1979), 263–274. MR553256
  15. G. Valiron, ‘Sur la dérivée des fonctions algéroïdes’, Bull. Soc. Math. France 59 (1931), 17–39. MR1504970
  16. J. von Rieth, Untersuchungen gewisser Klassen gewöhnlicher Differentialgleichungen erster und zweiter Ordnung im Komplexen (Ph.D. Thesis, Technischen Hochschule, Aachen, 1986).
  17. K. Yosida, ‘A generalization of Malmquist's theorem’, Japan J. Math. 9 (1933), 253–256.
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