A comparison theorem
Authors:
Walter Leighton and William Oo Kian Ke
Journal:
Proc. Amer. Math. Soc. 28 (1971), 185188
MSC:
Primary 34.42
DOI:
https://doi.org/10.1090/S00029939197102731216
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
Keywords:
Secondorder linear differential equation,
conjugate point,
comparison theorem
Article copyright:
© Copyright 1971
American Mathematical Society