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Uniform asymptotic solution of second order linear differential equations without turning varieties


Author: Gilbert Stengle
Journal: Math. Comp. 23 (1969), 1-22
MSC: Primary 34.50
DOI: https://doi.org/10.1090/S0025-5718-1969-0247197-5
MathSciNet review: 0247197
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Abstract: The purpose of this paper is to initiate the study of a new kind of asymptotic series expansion for solutions of differential equations containing a parameter. We obtain uniform asymptotic solutions for certain equations of the form

$\displaystyle { \in ^{2n}}y'' = a(t, \in )y,{\text{ }}({\text{ }})' = d/dt,$

where $ n$ is a positive integer, $ t$ and $ \in $ are real variables ranging over $ \left\vert t \right\vert \leqq {t_0},0 < \in \leqq { \in _0}$, and $ a$ is a function infinitely differentiable on the closure of this domain. We require that $ a(t, \in )$ satisfy conditions which can be regarded as generalized nonturning-point conditions. These conditions imply the absence of secondary turning points, and reduce in the simplest case to the condition $ a(t,0) \ne 0$, but also include cases (the interesting ones) in which $ a(0,0) = 0$

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1969-0247197-5
Article copyright: © Copyright 1969 American Mathematical Society

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