Annali di Matematica Pura ed Applicata

, Volume 81, Issue 1, pp 249–258 | Cite as

Sequences of nonlinear differential equations with related solutions

  • Mostafa A. Abdelkader


A second-order nonlinear differential equation which occurs (together with variants of it) in many problems of applied mathematics, physics and engineering is here reduced to a first-order equation. This equation contains a parameter which is a quadratic rational function of two parameters appearing in the original equation. By applying a certain identity for a quadratic rational function, two (finite or infinite) sequences of nonlinear differential equations are generated whose solutions are determinable whenever the solution of any equation belonging to a sequence is known. The cases amenable to exact solution by quadrature are given.


Differential Equation Exact Solution Rational Function Applied Mathematic Original Equation 
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  1. [1]
    E. A. Kearsley, Jour. Res. Nat. Bur. Stand. 67 B, p. 245–247 (1963).MathSciNetGoogle Scholar
  2. [2]
    A. C. Pipkin, Phys. Fluids 6, p. 1382–1388 (1963).CrossRefMATHMathSciNetGoogle Scholar
  3. [3]
    A. Melis andF. Testa, Rend. Sem. Fac. Sci. Univ. Cagliari 33, p. 438–444 (1963).MathSciNetMATHGoogle Scholar
  4. [4]
    R. Bellman,Stability Theory of Differential Equations, New York 1953, McGraw-Hill, Ch. VII.MATHGoogle Scholar
  5. [5]
    A. B. Pifko andM. A. Goldberg, AIAA Jour. 2, p. 1340–1342 (1964).CrossRefGoogle Scholar
  6. [6]
    P. A. Lindsay, Jour. Electronics Control 6, p. 415–431 (1959).MathSciNetGoogle Scholar
  7. [7]
    G. Ciobanu andI. Popescu, Jour. Electronics Control 16, p. 59–64 (1964).Google Scholar
  8. [8]
    L. Marton (Edit.),Advances in Electronics and Electron Physics, vol. 10, New York 1958, Academic Press.MATHGoogle Scholar
  9. [9]
    C. W. Jones, Proc. Roy. Soc. 217, p. 327–343 (1953).MATHGoogle Scholar
  10. [10]
    M. A. Abdelkader, SIAM Rev. 8, p. 526–529 (1966).CrossRefGoogle Scholar
  11. [11]
    S. G. Alikhanov, V. E. Zakharov, andG. L. Khorasanov, Plasma Phys. (Jour. Nuclear Energy Part C) 5, p 309–313 (1963).CrossRefGoogle Scholar
  12. [12]
    M. A. Abdelkader, Proc. Cambridge Phil. Soc. 64, p. 105–112 (1968).MATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    —— ——, Intern. Jour. Electronics 23, p. 449–471 (1967).Google Scholar
  14. [14]
    P. R. Stein andS. M. Ulam, Rozprawy Mat. 39, p. 1–66 (1964).MathSciNetGoogle Scholar

Copyright information

© Nicola Zanichelli Editore 1969

Authors and Affiliations

  • Mostafa A. Abdelkader

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