A class of second-order differential equations
The authors give new results on the qualitative behavior of solutions based on their study of the asymptotic behavior of solutions to the second-order linear differential equations \[ (pr')'+qr={c^2\over pr^3} \tag{a} \] and \[ (py')'+qy=0, \tag{b} \] where \(p,q\) are continuous, generally complex-valued, in an interval under consideration, \(p\neq 0\), and \(c\) is a nonzero constant. Five theorems and several corollaries concerning equations (a) and (b) are presented. The first statement (Theorem 1) develops a connection between these two equations. The next statement reads as follows. Theorem 2: Let \(p,q>0\) be real-valued, \(c>0\) and \[ \int_0^\infty \frac{ds}{p(s)}=\infty,\qquad \limsup_{t\to\infty} \root{4}\of{p(t)q(t)} \int_0^t \frac{ds}{p(s)}=\infty. \tag{1} \] Then any solution \(r\) to (a) satisfies \(\limsup_{t\to\infty}r(t)\root{4}\of {p(t)q(t)}\geq \sqrt c\). From theorems 1 and 2 it follows: If (1) holds, then equation (b) cannot have two linearly independent solutions \(u\), \(v\) both of which are \(o((pq)^{-1/4})\) as \(t\to\infty\). Further results are specialized to the case \(p=1\), \(q>0\), under the assumption that (b) is oscillatory. Theorem 3 estimates a product of solutions to equation (b). This gives a condition on \(q\) under which the equation \(y''+qy=0\) cannot have two linearly independent solutions \(u\), \(v\) such that \(u=O(q^{-1/4})\) and \(v=o(q^{-1/4})\). Theorems 4 and 5 describe the sequence of the extrema of \(y=(pq)^{1/4}r\), where \(r\) is a solution to (a). Such results are connected with a theorem of Wintner referred e.g. by \textit{P. Hartman} [Ordinary differential equations, New York. John Wiley \& Sons, Inc. (1964; Zbl 0125.32102)]. A general background for some results of the paper can be found in O. Bor ůvka [Linear differential transformations of the second order, Oxford Press, London (1971; Zbl 0222.34002)].
linear second-order differential equation
scientific article; zbMATH DE number 1310602
scientific article