Table of Contents Author Guidelines Submit a Manuscript
International Journal of Differential Equations
Volume 2010, Article ID 436860, 13 pages
http://dx.doi.org/10.1155/2010/436860
Research Article

Exact Solutions for Certain Nonlinear Autonomous Ordinary Differential Equations of the Second Order and Families of Two-Dimensional Autonomous Systems

Department of Engineering Sciences, University of Patras, 26504 Patras, Greece

Received 19 August 2010; Revised 22 November 2010; Accepted 30 November 2010

Academic Editor: Peiguang Wang

Copyright © 2010 M. P. Markakis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. G. Duffing, “Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz,” Braunschweig, vol. 134, 1918. View at Google Scholar
  2. B. van der Pol, “On relaxation oscillations,” Philosophical Magazine, vol. 2, no. 7, pp. 978–992, 1926. View at Google Scholar
  3. L. Rayleigh, “On maintained vibrations,” Philosophical Magazine, vol. 15, no. 1, article 229, 1883, see the "Theory of Sound", Dover, New York, NY, USA, 1945. View at Google Scholar
  4. S. Lynch, “Small amplitude limit cycles of the generalized mixed Rayleigh-Liénard oscillator,” Journal of Sound and Vibration, vol. 178, no. 5, pp. 615–620, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. A. Lins, W. de Melo, and C.C. Pugh, On Liénard’s Equation, vol. 597 of Lecture Notes in Mathematics, 1977.
  6. F. Dumortier and C. Li, “On the uniqueness of limit cycles surrounding one or more singularities for Liénard equations,” Nonlinearity, vol. 9, no. 6, pp. 1489–1500, 1996. View at Google Scholar · View at Scopus
  7. R. E. Mickens, “Liénard systems, limit cycles, melnikov theory, and the method of slowly varying amplitude and phase,” Journal of Sound and Vibration, vol. 217, no. 4, pp. 790–793, 1998. View at Google Scholar · View at Scopus
  8. S. Lynch, “Limit cycles of generalized Liénard equations,” Applied Mathematics Letters, vol. 8, no. 6, pp. 15–17, 1995. View at Google Scholar · View at Scopus
  9. S. Lynch, “Generalized cubic Liénard equations,” Applied Mathematics Letters, vol. 12, no. 2, pp. 1–6, 1999. View at Google Scholar · View at Scopus
  10. J. Garcia-Margallo and J. D. Bejarano, “The limit cycles of the generalized Rayleigh-Liénard oscillator,” Journal of Sound and Vibration, vol. 156, no. 2, pp. 283–301, 1992. View at Google Scholar · View at Scopus
  11. S. Lynch and C. J. Christopher, “Limit cycles in highly non-linear differential equations,” Journal of Sound and Vibration, vol. 224, no. 3, pp. 505–517, 1999. View at Google Scholar · View at Scopus
  12. R. E. Kooij and C. J. Christopher, “Algebraic invariant curves and the integrability of polynomial systems,” Applied Mathematics Letters, vol. 6, no. 4, pp. 51–53, 1993. View at Google Scholar · View at Scopus
  13. H. H. Denman, “Approximate invariants and lagrangians for autonomous, weakly non-linear systems,” International Journal of Non-Linear Mechanics, vol. 29, no. 3, pp. 409–419, 1994. View at Google Scholar · View at Scopus
  14. W. T. van Horssen, “Perturbation method based on integrating factors,” SIAM Journal on Applied Mathematics, vol. 59, no. 4, pp. 1427–1443, 1999. View at Google Scholar · View at Scopus
  15. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer, Berlin, Germany, 2nd edition, 1996.
  16. E. K. Ifantis, “Analytic solutions for nonlinear differential equations,” Journal of Mathematical Analysis and Applications, vol. 124, no. 2, pp. 339–380, 1987. View at Google Scholar · View at Scopus
  17. E. N. Petropoulou and P. D. Siafarikas, “Analytic solutions of some non-linear ordinary differential equations,” Dynamic Systems and Applications, vol. 13, no. 2, pp. 283–316, 2004. View at Google Scholar · View at Scopus
  18. I. Langmuir and K. B. Blodgett, “Currents limited by space charge between coaxial cylinders,” Physical Review, vol. 22, no. 4, pp. 347–356, 1923. View at Publisher · View at Google Scholar · View at Scopus
  19. H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, New York, NY, USA, 1962.
  20. A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2nd edition, 2003.
  21. M. P. Markakis, “Closed-form solutions of certain Abel equations of the first kind,” Applied Mathematics Letters, vol. 22, no. 9, pp. 1401–1405, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  22. E. Kamke, Losungmethoden und Losungen, B.G. Teubner, Stuttgart, Germany, 1983.