On some continuities of phases theory and WKB method for linear differential equation of the second order. (English) Zbl 0752.34034

Summary: It is found the necessary and sufficient condition for having the first phase of the differential equation \(y''+q(t)y=0\) in the form \(\alpha(t)=\varphi\left[\int^ t\sqrt{q(\tau)}d\tau\right]\), where \(\varphi=\varphi(s)\in C^ 3\), \(\varphi'(s)\neq 0\) is a given function. This problem is solved in a connection with the case when the series in the exponent of the WKB solution of the given equation with a suitable coefficient \(q(t)\) reduces into a finite sum.


34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34A30 Linear ordinary differential equations and systems
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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[1] Borůvka O.: Lineare Differentialtransformationen 2.Ordnung. VEB Deutscher Verlag der Wissenschaften, Berlin 1967. · Zbl 0153.11201
[2] Fedoryuk M.V.: Asymptotic Methods for Linear Differential Equations. (Russian), Nauka, Moscow, 1983. · Zbl 0538.34001
[3] Hartman P.: Ordinary Differential Equations. (Russian), Mir, Moscow, 1970. · Zbl 0214.09101
[4] Laitoch M.: Homogene lineare zu sich selbst begleitende Differentialgleichung zweiter Ordnung. Acta UPO, Fac. rer. nat., Tom. 33, Math. X (1971), 61-72. · Zbl 0298.34006
[5] Kamke E.: Differentialgleichungen. Lösungsmethoden und Lösungen. Leipzig, 1959. · Zbl 0026.31801
[6] Zeman J.: Über eine Anwendung der Phasentheorie. Acta UPO, Fac.rer.nat., Tom. 53, Math. XVI (1977), 137-140. · Zbl 0415.34015
[7] Zeman J.: Eine Bemerkung zur Methode WKB. Acta UPO, Fac.rer.nat., Tom. 57, Math. XVII (1978), 61-68. · Zbl 0457.34045
[8] Zeman J.: Zur asymptotischen Integration der Differentialgleichung y” + q(t)y = 0. Acta UPO, Fac.rer.nat., Tom 69, Math. XX (1981), 129-132. · Zbl 0487.34062
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