Conditions for solvability of the Hartman-Wintner problem in terms of coefficients

The Equation (1) (r(x)y′)′ = q(x)y(x) is regarded as a perturbation of (2) (r(x)z′(x))′ = q₁(x)z(x). The functions r(x), q₁(x) are assumed to be continuous real valued, r(x) > 0, q₁(x) ≥ 0, whereas q(x) is continuous complex valued. A problem of Hartman and Wintner regarding the asymptotic integration of (1) for large x by means of solutions of (2) is studied. Sufficiency conditions for solvability of this problem expressed by means of coefficients r(x), q(x), q₁(x) of Equations (1) and (2) are obtained.