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Common increasing dispersions of certain linear second order differential equations. (English) Zbl 0615.34041

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 \(L^+_ q\subset L^+_ p\).

MSC:

34C99 Qualitative theory for ordinary differential equations
34A30 Linear ordinary differential equations and systems

Citations:

Zbl 0222.34002
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References:

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