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


34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34A30 Linear ordinary differential equations and systems
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[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
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