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Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations)

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  • Dimitrios E. Panayotounakos
  • Theodoros I. Zarmpoutis

Abstract

We provide a new mathematical technique leading to the construction of the exact parametric or closed form solutions of the classes of Abel's nonlinear differential equations (ODEs) of the first kind. These solutions are given implicitly in terms of Bessel functions of the first and the second kind (Neumann functions), as well as of the free member of the considered ODE; the parameter 𠜈 being introduced furnishes the order of the above Bessel functions and defines also the desired solutions of the considered ODE as one-parameter family of surfaces. The nonlinear initial or boundary value problems are also investigated. Finally, introducing a relative mathematical methodology, we construct the exact parametric or closed form solutions for several degenerate Abel's equation of the first kind.

Suggested Citation

  • Dimitrios E. Panayotounakos & Theodoros I. Zarmpoutis, 2011. "Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations)," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2011, pages 1-13, October.
    • Handle: RePEc:hin:jijmms:387429
    DOI: 10.1155/2011/387429
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    Cited by:

    1. Pun, Chi Seng, 2018. "Time-consistent mean-variance portfolio selection with only risky assets," Economic Modelling, Elsevier, vol. 75(C), pages 281-292.
    2. Carmen Ionescu & Radu Constantinescu, 2022. "Solving Nonlinear Second-Order Differential Equations through the Attached Flow Method," Mathematics, MDPI, vol. 10(15), pages 1-14, August.
    3. Yutaka Okabe & Akira Shudo, 2020. "A Mathematical Model of Epidemics—A Tutorial for Students," Mathematics, MDPI, vol. 8(7), pages 1-16, July.

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