Singular Cauchy Problem for an Ordinary Differential Equation Unsolved with Respect to the Derivative of the Unknown Function

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For a singular Cauchy problem

$$\sum \limits_{i=0}^N\sum \limits_{j=0}^N\sum \limits_{k=0}^N{a}_{ijk}{t}^i{\left(x(t)\right)}^j{\left({x}^{\prime }(t)\right)}^k+\varphi \left(t,x(t),{x}^{\prime }(t)\right)=0\kern0.5em x(0)=0,$$

where N ≥ 2 and a ijk are constants, a00k = 0, k ∈ {0, 1, . . .,N} , a100 ≠ 0, a010 ≠ 0, a ijk = 0, 1 ≤ i + j < m, k ∈ {1, . . . ,N} , 2 ≤ mN, and φ is a function small in a certain sense, we find a nonempty set of continuously differentiable solutions x: (0, ρ] → ℝ, where ρ is sufficiently small, such that

$${\displaystyle \begin{array}{cc}x(t)=\sum \limits_{k=1}^m{c}_k{t}^k+o\left({t}^m\right),& t\to +0,\end{array}}$$

where c1, . . . , c m are known constants.

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Translated from Neliniini Kolyvannya, Vol. 20, No. 2, pp. 166–183, April–June, 2017.

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Zernov, A.E., Kuzina, Y.V. Singular Cauchy Problem for an Ordinary Differential Equation Unsolved with Respect to the Derivative of the Unknown Function. J Math Sci 231, 712–729 (2018). https://doi.org/10.1007/s10958-018-3846-5