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

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, a_00k = 0, k ∈ {0, 1, . . .,N} , a₁₀₀ ≠ 0, a₀₁₀ ≠ 0, a _ijk = 0, 1 ≤ i + j < m, k ∈ {1, . . . ,N} , 2 ≤ m ≤ N, 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.