Annali di Matematica Pura ed Applicata

, Volume 117, Issue 1, pp 339–347 | Cite as

Analytical theory of non-linear oscillations (+)

VIII. — Second order conservative systems whose solutions are all oscillating with the least period 2π
  • Chike Obi


This paper establishes what are essentially necessary and sufficient conditions under which every solution of the differential equation\(\ddot x + g\left( x \right) = 0\left( {\dot x = dx/dt} \right)\) in the real domain oscillates with the least period 2π independent of the initial conditions of the solution and in addition gives expressions for the Fourier coefficients of a solution with given initial conditions.


Differential Equation Analytical Theory Fourier Coefficient Real Domain 
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  1. [1]
    I. Koukles -N. Piskounov,Sur les vibrations tautochrones dans les systèmes conservatifs et non conservatifs, C. R. Acad. Sci. URSS,17 (1937), pp. 471–475.Google Scholar
  2. [2]
    J. J. Levin -S. S. Schatz,Non-linear oscillations of fixed period, J. Math. Anal. Appl.,7 (1973), pp. 284–288.CrossRefGoogle Scholar
  3. [3]
    C. Obi,Analytic theory of non-linear oscilations. VII: The periods of the periodic solutions of the equation\(\ddot x + g\left( x \right) = 0\), J. Math. Anal. Appl.,55 (1976), pp. 295–301.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    C. Obi,Periodic solutions of non-linear differential equations of the second order IV, Proc. Cambridge Philos. Soc.,47 (1951), pp. 741–751.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    C. Obi,Analytical theory of non-linear oscillations. IX: Limit oscillations and unbounded solutions of the equation\(\ddot x + \lambda ^2 g\left( x \right) = \} \) cos ωt, Ann. Mat. Pura Appl. (to appear).Google Scholar
  6. [6]
    C. Obi, Equations of the form\(\ddot x + g\left( x \right) = 0\) with periodic solutions whose Fourier series are finite, Nigerian Journal of Science (to appear).Google Scholar
  7. [7]
    M. Urabe,Potential forces which yield periodic motions of fixed period, J. Math. Mech.,10 (1961), pp. 569–578.zbMATHMathSciNetGoogle Scholar
  8. [8]
    M. Urabe, Relations between periodics and amplitudes of periodic solutions of\(\ddot x + g\left( x \right) = 0\), Funkcialaj Ekvacioj,6 (1964), pp. 63–88.zbMATHMathSciNetGoogle Scholar
  9. [9]
    E. T. Whittaker - G. N. Watson,Modern Analysis, Cambridge (1946).Google Scholar

Copyright information

© Nicola Zanichelli Editore 1978

Authors and Affiliations

  • Chike Obi
    • 1
  1. 1.LagosNigeria

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