## 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

### Keywords:

oscillatory equation; dispersion
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### 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
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