×

On a structure of the intersection of the set of dispersions of two second-order linear differential equations. (English) Zbl 0543.34026

A function \(X\in C^ 3(j)\), X’(t)\(\neq 0\) for \(t\in j:=(a,b)\subset {\mathbb{R}}\) is said to be a dispersion of (q) \(y''=q(t)y q\in C^ 0({\mathbb{R}})\) if it is a solution of the differential equation \[ (- 1/2)X'''/X'+(3/4)(X''/X')^ 2+X^{'2}\cdot q(X)=q(t). \] Denote \(L_ q\) the set of dispersions of (q). Let \(q_ 1\in C^ 0({\mathbb{R}})\), \((q_ 1/q_ 2)\in C^ 2({\mathbb{R}}),\quad q_ 1(t)\neq q_ 2(t)\) for \(t\in {\mathbb{R}}\) and let \(q_ 1\) be oscillatory. In this paper the algebraic structure of the set \(L_{q_ 1}\cap L_{q_ 2}\) is investigated. New results are obtained owing to lectures at the seminar of the Institute of Mathematics of the Czechoslovak Academy of Science in Brno given by Prof. Boruvka.
Reviewer: M.Hačik

MSC:

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34A30 Linear ordinary differential equations and systems
PDF BibTeX XML Cite
Full Text: EuDML

References:

[1] O. Borůvka: Linear Differential Transformations of the Second Order. The English Univ. Press, London, 1971. · Zbl 0218.34005
[2] О. Борувка: Тєоруя глобальных свойсмв обыкновєнных лунєйных дуффєрєнцуальных ураєнєный вморого порядка. Дифференциальные уравнения, No 8, t. XII, 1976, 1347-1383.
[3] O. Borůvka: Lectures at the seminar of the Institute of Mathematics of the Czechoslovak Academy of Science in Brno. · Zbl 0218.34005
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.