Differential Equations

  • Igor Nikolaev
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 41)


Ordinary differential equations (ODE) correspond to the orientable foliations. Allowing the ODE to be unsolved with respect to the derivative of highest order gives us an interesting class of non-orientable foliations. A method of integration of such differential equations has been suggested by Cayley and Darboux in the context of principal curvature lines on surfaces. Later Hartman and Wintner in the series of works [138] — [140] developed a general method of the integration of such ODE’s. In this chapter we introduce the reader to the theory of Hartman and Wintner as well as to the later (geometric) method of A. G. Kuzmin.


Singular Point Simple Root Integral Curve Local Scheme Ordinary Differential Equation 
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Bibliographic Notes

  1. 239.
    Picard, É, 1895 Sur les points singuliers des équations différentielles du premier ordre, Math. Annalen. 46, 521–528.MathSciNetMATHCrossRefGoogle Scholar
  2. 138.
    Hartman, P. and Wintner, A., 1953 On the behavior of the solutions of real binary differential systems at singular points, Amer. J. of Math. 75, 117–126.MathSciNetMATHCrossRefGoogle Scholar
  3. 139.
    Hartman, P. and Wintner, A., 1953 Envelopes and discriminant curves, Amer. J. of Math. 75, 142–158.MathSciNetMATHCrossRefGoogle Scholar
  4. 140.
    Hartman, P. and Wintner, A., 1953 On the singularities in nets of curves defined by differential equations, Amer. J. of Math. 75, 277–297.MathSciNetMATHCrossRefGoogle Scholar
  5. 173.
    Kuzmin, A. G., 1982 The behavior of the integral curves in a neighborhood of a branch point of the discriminant curve, Vestnik Leningrad Univ. Math 14, 143–149.Google Scholar
  6. 209.
    Nemytskii V. V. and Stepanov V. V., 1960 Qualitative Theory of Differential Equations, Princeton Univ. Press.MATHGoogle Scholar
  7. 174.
    Kuzmin, A. G., 1982 Indices of singular and irregular points of direction fields in the plane, Vestnik Leningrad Univ. Math 14, 285–292.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Igor Nikolaev
    • 1
  1. 1.The Fields Institute for Research in Mathematical SciencesTorontoCanada

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