The author considers oscillatory properties of \[ \text{(H)}\quad [p(x) y'(x)]'+ q(x) y(x)= 0 \qquad \text{and\qquad (NH)}\quad [p(x) y'(x)]'+ q(x) y(x)= f(x). \] In introducing the Kummer-Liouville transformation, the author makes a misleading statement \(y(x)= w(x) z(x)\). The classical representation is: \(y(x)= w(x) y(\xi)\), where \(\xi= \varphi(x)\), where \(\varphi(x)= \int^ x [p(\theta) w^ 2(\theta)]^{-1}d\theta\) is the usual choice (but not a necessary choice). Then the original equation (H) is duplicated: \((\text{H}')\) \((R(\xi)z'(\xi)'+ Q(\xi) z(\xi)=0\), where \(R(\xi)= R(\xi(x))= p(x)\varphi'(x) w^ 2(x)\), \(Q(\xi)= Q(\xi(x))= (p(x) w'(x))'+ q(x) w(x)[\varphi'(x)]^{-1} w(x)\). Instead, the author produces an incorrect version (equation 0.2).
It has been known that the product \(pq\) has important physical interpretation (product of inductance and capacitance in the electrical interpretation of (H)). Here, the author establishes a connection between boundedness of solutions of (H) and integrability: \(\int^ p(pq)'(py')^ 2/(pq)^ 2 dx\).
Theorem 2.4 is a trivial consequence of properties of the Wronskian. You can not have two solutions, where one is oscillatory and the other one is not.
Theorem 2.7 stating that if (NH) equation has a solution \(r(x)<0\) then the equation (H) is disconjugate is clearly false. Consider \(p(x)\equiv 1\), \(q(x)\equiv 1\), and (NH) equation \(y''+ y={1\over 2}\), \(y\equiv- {1\over 2}\) solves it. The author did not fully state all conditions of the intended theorem. There are some good ideas in his proposed techniques, but in the present form this article should not have been published.