Annali di Matematica Pura ed Applicata

, Volume 107, Issue 1, pp 373–381 | Cite as

On approximating conjugate, focal, and σ-points for linear differential equations of second order

  • Walter Leighton


A study of the fundamental nature of solving a system of n ordinary (nondifferential) equations is undertaken in § 1. It is found that the properties of the inverse function play an unexpectedly decisive rôle. In § 2, the results of § 1 are applied to methods of approximating eigenvalues, conjugate, focal, and σ-points previously introduced by the author. Finally, the connection between these methods and Wiman's integral is indicated.


Differential Equation Inverse Function Linear Differential Equation Fundamental Nature 
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  1. [1]
    P. Hartman -A. Wintner,The asymptotic arcus variation of solutions of real linear differential equations of second order, American Journal of Mathematics,70 (1948), pp. 1–10.MathSciNetGoogle Scholar
  2. [2]
    W. Leighton,Upper and lower bounds for eigenvalues, Journal of Mathematical Analysis and Applications,35 (1971), pp. 381–388.CrossRefMATHMathSciNetGoogle Scholar
  3. [3]
    W. Leighton,Computing bounds for focal points and for σ-points for second-order linear differential equations, Ordinary Differential Equations, Academic Press, New York (1972).Google Scholar
  4. [4]
    A. Wiman,Über die reellen Lösungen der linearen Differentialgleichungen zweiter Ordnung, Arkif für Matematik, Astronomi och Fysik,12 (1917), pp. 1–22.Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1975

Authors and Affiliations

  • Walter Leighton
    • 1
  1. 1.ColumbiaU.S.A.

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