Asymptotic integration of a second order ordinary differential equation
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- by Jaromír Šimša PDF
- Proc. Amer. Math. Soc. 101 (1987), 96-100 Request permission
Abstract:
Equation (1) $(r(t)x’)’ + f(t)x = 0$ is regarded as a perturbation of (2) $(r(t)y’)’ + g(t)y = 0$, where the latter is nonoscillatory at infinity. It is shown that if a certain improper integral involving $f - g$ converges sufficiently rapidly (but perhaps conditionally), then (1) has a solution which behaves for large $t$ like a principal solution of (2). The proof of this result is presented in such a way that it also yields as a by-product an improvement on a recent related result of Trench.References
- Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
- William F. Trench, Linear perturbations of a nonoscillatory second order equation, Proc. Amer. Math. Soc. 97 (1986), no. 3, 423–428. MR 840623, DOI 10.1090/S0002-9939-1986-0840623-1
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 101 (1987), 96-100
- MSC: Primary 34E10; Secondary 34C10
- DOI: https://doi.org/10.1090/S0002-9939-1987-0897077-X
- MathSciNet review: 897077