Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Instability intervals and growth rates for Hill's equation


Author: Joseph B. Keller
Journal: Quart. Appl. Math. 66 (2008), 191-195
MSC (2000): Primary 34B30
DOI: https://doi.org/10.1090/S0033-569X-07-01083-1
Published electronically: December 5, 2007
MathSciNet review: 2396657
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Abstract | References | Similar Articles | Additional Information

Abstract: Hill's equation is a real linear second-order ordinary differential equation with a periodic coefficient $ f(t)$:

$\displaystyle y^{\prime\prime} (t) +\left[ \lambda+\varepsilon f\left(t\right) \right] y(t) =0.$ (0.1)

It has unbounded solutions for certain intervals of the real parameter $ \lambda$, called instability intervals. Here these intervals, and the growth rate of the unbounded solutions, are determined for $ \varepsilon$ small, and also for $ \lambda$ large. This is done by constructing a fundamental pair of solutions which are power series in $ \varepsilon/\lambda^{1/2}$, with coefficients that are bounded functions of $ \lambda$.


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Additional Information

Joseph B. Keller
Affiliation: Departments of Mathematics and Mechanical Engineering, Stanford University, Stanford, California 94305-2125
Email: keller@math.stanford.edu

DOI: https://doi.org/10.1090/S0033-569X-07-01083-1
Received by editor(s): June 28, 2007
Published electronically: December 5, 2007
Article copyright: © Copyright 2007 Brown University
The copyright for this article reverts to public domain 28 years after publication.


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