Staněk, Svatoslav Common increasing dispersions of certain linear second order differential equations. (English) Zbl 0615.34041 Acta Univ. Palacki. Olomuc., Fac. Rerum Nat. 82, Math. 24, 51-60 (1985). A function \(X\in C^ 3(R)\), \(X'(t)>0\) for \(t\in R\), is called an increasing (complete) dispersion of equation (q): \(y''=q(t)y\) \((q\in C^ 0(R))\) if \(X(R)=R\) and X is a solution of the Kummer differential equation \[ -X\prime''/2X'+3/4(X''/X')^ 2+X^{'2}q(X)=q(t) \] [see O. Borůvka, Linear differential transformations of the second order (1971; Zbl 0222.34002)]. The set of increasing dispersions of (q) forms a group \(L^+_ q\) relative to the composition of functions. Let (q) be a disconjugate (pure disconjugate or specially disconjugate) equation. This paper describes by means of the phase theory all equations of the type (p): \(y''=p(t)y\) \((p\in C^ 0(R))\) which are disconjugate or oscillatory such that \(L^+_ q\subset L^+_ p\). Cited in 1 Document MSC: 34C99 Qualitative theory for ordinary differential equations 34A30 Linear ordinary differential equations and systems Keywords:second order differential equation; dispersion; Kummer differential equation; increasing dispersions; phase theory Citations:Zbl 0222.34002 PDFBibTeX XMLCite \textit{S. Staněk}, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 24, 51--60 (1985; Zbl 0615.34041) Full Text: EuDML References: [1] Blanton G., Baker J. A.: Iteration groups generated by \(C^\ast\) functions. Arch. Math. (Brno) 3, XVIII, 1982, 121-128. · Zbl 0518.26002 [2] Borůvka O.: Linear Differential Transformations of the Second Order. The English Universities Press, London, 1971. · Zbl 0218.34005 [3] Борувка О.: Тєория глобальных свойств обыкновєнных лунєйных диффєрєнциальных уравнєний второго порядка. Диффєрєнциальныє уравнєния, Но. 8, t. 12, 1976, 1347-1383. [4] Borůvka O.: Sur les Transformations Simultanées de Deux Équations Différentielles Linéaires du Deuxiéme Order dans Elles-mênes. Applicable Analysis, 1983, No 1-4, 187-200. [5] Borůvka O.: Lectures at the seminar of the institute of Mathematics of the Czechoslovak Academy of Science in Brno. · Zbl 0218.34005 [6] Staněk S.: On a structure of the intersection of the set of dispersions of two second-order linear differential equations. Acta Univ. Palackianae Olomucensis F. R. N., 73, 1982, 79 - 85. · Zbl 0543.34026 [7] Staněk S.: On the intersection of the set of solutions of two Kummer’s differential equations. Acta Univ. Palackianae Olomucensis F. R. N., 79, 1984, 25-32. · Zbl 0596.34014 [8] Staněk S.: On the intersection of groups of dispersions of the equation \(y'' = q(t)y\) and its accompaying equation. Acta Univ. Palackianae Olomucensis F. R. N., 79, 1984, 33-38. · Zbl 0596.34015 [9] Staněk S.: On the intersection of groups of increasing dispersions in two second order oscillatory linear differential equations. Acta Univ. Palackianae Olomucensis F. R. N., 82, 1985, 61-66. · Zbl 0615.34042 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.