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Analytic integration of a system of nonlinear ordinary differential equations with an irregular type singularity. - II

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Summary

We shall study the same differential equations(A) as in our previous paper[5]. The equations(A) admit a formal solution of the form(F) which is expressed by a formal power series of x and φ(x), where φ(x) is a holomorphic general solution of a differential equation of the form(R). Under an additional assumption that the solution φ(x) vanishes identically when the arbitrary constants involved in it are equal to zero, it is already shown in[5] that the formal solution(F c), which is obtained by formally rearranging(F) in power series of φ(x), is uniformly convergent whenever the values of x and φ(x) satisfy inequalities of the form(0.14). The present paper discusses about how to study an analytic meaning of the formal solution(F) without the additional assumption mentioned above.

References

  1. [1]

    M. Hukuhara,Intégration formelle d'un système d'équations différentielles non linéaires dans le voisinage d'un point singulier, Ann. di Mat. pura e appl., (V),19 (1940), pp. 35–44.

  2. [2]

    M. Hukuhara,Sur les points singuliers des équations différentielles linéaires, III, Mem. Fac. Sci. Kyûsyû Imp. Univ.,2 (1942), pp. 125–137.

  3. [3]

    M. Hukuhara,Renzokuna Kansû no Zoku to Syazô, Mem. Fac. Sci. Kyûsyû Univ.,5 (1950), pp. 161–163;Sur l'existence des points invariants d'une transformation dans l'espace fonctionnel, Japanese Jour. Math.,20 (1950), pp. 1–4.

  4. [4]

    M. Iwano,Analytic expressions for bounded solutions of non-linear ordinary differential equations with an irregular type singular point, Ann. di Mat. pura e appl. (IV),82 (1969), pp. 189–256.

  5. [5]

    M. Iwano,Analytic integration of a system of nonlinear ordinary differential equations with an irregular type singularity, ibid.,94 (1972), pp. 109–160.

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Dedicated to Professor Hugh.L. Turrittin on the occasion of his retirement.

Entrata in Redazione il 27 luglio 1972.

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Iwano, M. Analytic integration of a system of nonlinear ordinary differential equations with an irregular type singularity. - II. Annali di Matematica 99, 221–246 (1974). https://doi.org/10.1007/BF02413727

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Keywords

  • Differential Equation
  • Ordinary Differential Equation
  • General Solution
  • Power Series
  • Additional Assumption