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First phases of the associated equation with parameters for the differential equation \(y''=q(t)y\). (English) Zbl 0852.34014

The paper deals with the second-order linear differential equation of Jacobi form (1) \(y''= q(t) y\), \(t\in (a, b)\), \(-\infty\leq a< b\leq + \infty\). The author considers the first amplitude \(r\), the second amplitude \(s\), the \(a\)-amplitude \(\sigma\) with parameters \([k, \lambda]\), the first phase \(\alpha\), the second phase \(\beta\), and the \(a\)-phase \(\gamma\) with parameters \([k, \lambda]\), where \(s(t)= \sqrt{u^2+ v^2}\), \(r(t)= \sqrt{u^{\prime 2}+ v^{\prime 2}}\), \(\sigma(t)= \sqrt{(ku+ \lambda u')^2+ (kv+ kv')^2}\), \(k, \lambda\in \mathbb{R}\), \(k^2+ \lambda^2> 0\), \(\tan \alpha(t)= u(t)/v(t)\), \(\tan \beta(t)= u'(t)/ v'(t)\), \(\tan \gamma(t)= (ku(t)+ \lambda u'(t))/(kv(t)+ \lambda v'(t))\), \((u(t), v(t))\) is the basis of (1). The properties of the amplitudes and the phases are investigated.
Reviewer: S.Mazanik (Minsk)

MSC:

34A30 Linear ordinary differential equations and systems
34B05 Linear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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References:

[1] Borůvka O.: Lineare Differentialtransformationen 2. Ordnung. VEB DVW, Berlin 1967. · Zbl 0153.11201
[2] Laitoch M.: L’équation associée dans la théorie des transformation differentielles du second ordre. Acta UP Olomucensis, Fac. rer. nat., T. 12, 1963, 45-62.
[3] Laitoch M.: Homogene lineare zu sich selbst begleitende Differentialgleichung zweiter Ordnung. Acta UP Olomucensis, Fac. rer. nat., T. 33, 1971, 61-72. · Zbl 0298.34006
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[5] Votava M.: N\?které vlastnosti druhé zobecn\?né fáze diferenciální rovnice 2. řádu Jacobiho typu. Acta facultas pedagogicae Ostraviensis, 101 (1986), Series A-21, 65-83.
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