SINGULAR CAUCHY PROBLEM FOR AN ORDINARY DIFFERENTIAL EQUATION UNSOLVED WITH RESPECT TO THE DERIVATIVE OF THE UNKNOWN FUNCTION

摘要

For a singular Cauchy problem ∑ i = 0 N ∑ j = 0 N ∑ k = 0 N a ijk t i x t j x ′ t k + φ t x t x ′ t = 0 x 0 = 0 , $$ sum limits_{i=0}^Nsum limits_{j=0}^Nsum limits_{k=0}^N{a}_{ijk}{t}^i{left(x(t)right)}^j{left({x}^{prime }(t)right)}^k+varphi left(t,x(t),{x}^{prime }(t)right)=0kern0.5em x(0)=0, $$ where N ≥ 2 and a ~( ijk )are constants, a ~(00 k )= 0, k ∈ {0, 1, . . ., N } , a ~(100)≠ 0, a ~(010)≠ 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 x t = ∑ k = 1 m c k t k + o t m , t → + 0 , $$ {displaystyle begin{array}{cc}x(t)=sum limits_{k=1}^m{c}_k{t}^k+oleft({t}^mright),& tto +0,end{array}} $$ where c ~(1), . . . , c ~( m )are known constants.