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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.

MSC:

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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References:

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