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Convergence or Divergence of Formal Solutions of First Order Singular Nonlinear Ordinary Differential Equations in Complex Domain

Hidetaka Usui
2010 Funkcialaj Ekvacioj  
We study the convergence or divergence of formal solutions of first order singular nonlinear ordinary di¤erential equation (SE) f ðx; u; u 0 Þ ¼ 0, uð0Þ ¼ 0; where the function f is holomorphic in a neighbourhood of the origin of C 3 . The equation (SE) is said to be singular if f ð0; 0; xÞ 1 0. M. Miyake and A. Shirai [2, 3] already studied this problem in case of partial di¤erential equation and they gave a criterion of convergence or divergence of formal solutions by using the Taylor
more » ... the Taylor coe‰cients of order 1 of formal solution and equation. In this paper, we give a criterion by using the Taylor coe‰cients of higher order of formal solution and equation.
doi:10.1619/fesi.53.89 fatcat:xsl6lue5nrb3pkwik4joecfj3m