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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A comparison theorem


Authors: Walter Leighton and William Oo Kian Ke
Journal: Proc. Amer. Math. Soc. 28 (1971), 185-188
MSC: Primary 34.42
MathSciNet review: 0273121
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Abstract: In this paper the authors consider a pair of differential equations $ {y''_1} + {p_1}(x){y_1} = 0,{y''_2} + {p_2}(x){y_2} = 0$, where $ {p_i}(x)$ are positive and continuous, and where solutions $ {y_1}(x)$ and $ {y_2}(x)$ have common consecutive zeros at $ x = a$ and $ x = b$. They show that if the curves $ y = {p_1}(x)$ and $ y = {p_2}(x)$ have a single intersection (possibly a closed subinterval) and if $ {p_1}(a) > {p_2}(a),{p_2}(b) > {p_1}(b)$, the first conjugate point of $ a + {\varepsilon }$ ( $ {\varepsilon } > 0$ and small) for the second equation precedes that of the first.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0273121-6
PII: S 0002-9939(1971)0273121-6
Keywords: Second-order linear differential equation, conjugate point, comparison theorem
Article copyright: © Copyright 1971 American Mathematical Society