Asymptotic formulas are obtained in the paper for x → +∞ for the fundamental system of solutions to the equation$$l(y): = {i^{2n + 1}}\{ {(q{y^{(n + 1)}})^{(n)}} + {(q{y^{(n)}})^{(n + 1)}}\} + py = \lambda y,\;\;\;\;\;\;x \in I: = [1, + \infty ),$$where λ is a complex parameter. It is assumed that q is a positive continuously differentiable function, p has the form p = σ^(k), 0 ≤ k ≤ n, where σ⁽ is a locally integrable on I function, and the derivative is understood in the sense of the theory of distributions. In the case when k = 0 and λ ≠ 0, and the coefficients q and p of the expression l(y) are such that q =1/2 + q₁, and q₁, σ(= p) are integrable on I, restrictions on q₁ and σ and for any 1 ≤ k ≤ n − 1. For k = n additional constraints arise on these functions. We consider separately the case when λ = 0.Asymptotic formulas were also obtained for solutions to the equation l(y) = λy under the condition \(q(x) = \alpha {x^{2n + 1 + \nu }}{(1 + r(x))^{ - 2}},\;\sigma (x) = {x^{k + \nu }}(\beta + s(x))\), where α ≠ 0 mid β are complex numbers, ν ⩾ 0, and the functions r and s satisfy certain conditions of integral decay.

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奇数线性微分方程解的渐近性

本文针对x →+∞为方程$$ l（y）的基本解系统获得了渐近公式：= {i ^ {2n + 1}} \ {{（q {y ^ {（n + 1）}}）^ {（（n）}} + {（q {y ^ {（n）}}）^ {（n + 1）}} \} + py = \ lambda y，\; \; \; \; \; \; x \ in I：= [1，+ \ infty），$$其中λ是一个复数参数。据推测，q是正的连续可微函数，p具有形式p = σ ^（ķ），0≤ ķ ≤ Ñ，其中σ ^（是局部可微的上余功能，和所述衍生物在的意义上理解在k = 0且λ≠0的情况下，系数q和p表达的升（Ý）使得q = 1/2 + Q ₁，和q ₁，σ（= p）上积我，上限制q ₁和σ和任何1≤ ķ ≤ ñ - 1.对于k = n，在这些函数上会出现其他约束。我们分别考虑λ = 0的情况。在条件下，方程l（y）= λy的解也得到了渐近公式\（q（x）= \ alpha {x ^ {2n +1 + \ nu}} {（1 + r（x））^ {-2}}，\; \ sigma（x）= {x ^ {k + \ nu}}（\ beta + s（x））\），其中α ≠0 midβ是复数ν⩾0，并且函数r和s满足积分衰减的某些条件。