Asymptotics of Solutions to Linear Differential Equations of Odd Order

Asymptotic formulas are obtained in the paper for x -> +infinity for the fundamental system of solutions to the equation l(y) := i(2n+1) {(qy((n+1)))((n)) + (qy((n)))((n+1))} + py = lambda y, x is an element of I := [1, +infinity), where lambda is a complex parameter. It is assumed that q is a positive continuously differentiable function, p has the form p = sigma((k)), 0 <= k <= n, where sigma 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 lambda not equal 0, and the coefficients q and p of the expression l(y) are such that q = 1/2 + q(1), and q(1) , sigma(= p) are integrable on I, these formulas are well known. It was established in the paper that they are valid under the same restrictions on q(1) and sigma and for any 1 <= k <= n - 1. For k = n additional constraints arise on these functions. We consider separately the case when lambda = 0. Asymptotic formulas were also obtained for solutions to the equation l(y) = lambda y under the condition q(x) = alpha x(2n+1+nu)(1+r(x))(-2), sigma(x) = x(k+nu)(beta+s(x)), where alpha not equal 0 and beta are complex numbers, v >= 0, and the functions r and s satisfy certain conditions of integral decay.