Asymptotics of Solutions to Linear Differential Equations of Odd Order

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.