Abstract
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.
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Bibliographic Notes
Picard, É, 1895 Sur les points singuliers des équations différentielles du premier ordre, Math. Annalen. 46, 521–528.
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.
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Hartman, P. and Wintner, A., 1953 On the singularities in nets of curves defined by differential equations, Amer. J. of Math. 75, 277–297.
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.
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Kuzmin, A. G., 1982 Indices of singular and irregular points of direction fields in the plane, Vestnik Leningrad Univ. Math 14, 285–292.
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© 2001 Springer-Verlag Berlin Heidelberg
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Nikolaev, I. (2001). Differential Equations. In: Foliations on Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 41. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04524-4_15
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DOI: https://doi.org/10.1007/978-3-662-04524-4_15
Publisher Name: Springer, Berlin, Heidelberg
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