Asymptotic integration of linear ordinary differential equations of order \(N\) (Q1288900)

The authors discuss the existence of the basis of special form for the space of solutions to the perturbed \(n\)th-order linear ODE: \(M[x]=N[x]\) with: \[ M[y]=x^{(n)}+a_1(t)x^{(n-1)}+ \dots +a_n (t)x,\quad N[x]=b_p(t)x^{(p)}+ \dots +b_0(t)x, \] with \(0 \leq r \leq p \leq n-1\), \(1\leq i \leq n \). This basis has to be of the following form: \[ y^{(r)}_i=(1+\rho^r_i)x^{(r)}, \] where the functions \(\rho ^r_i\) are small at \(t=\infty\), and \(x_1\), \(x_2\), \dots, \(x_n\) are elements of the basis for the nonperturbed equation: \[ M[x]=0. \] An analogue problem can be stated for systems of ODEs of first order and this problem was already investigated by several authors. However, the theory created for first order systems does not apply to the single \(n\)th-order equation in an optimal way. This paper contains a theory created specially for a single \(n\)th-order equation. The authors prove that the existence of the basis of desired character depends on the so-called scalar Levinson dichotomy in the nonperturbed equation. Special sections of this paper are devoted to operators with constant coefficients, and to the so-called singular case, that is, the case, when the characteristic polynomial of the equations has zero roots.