Asymptotic integrations of nonoscillatory second order differential equations
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- by Shao Zhu Chen PDF
- Trans. Amer. Math. Soc. 327 (1991), 853-865 Request permission
Abstract:
The linear differential equation (1) $(r(t)x’)’ + (f(t) + q(t))x= 0$ is viewed as a perturbation of the equation (2) $(r(t)y’)’ + (f(t)y = 0$, where $r > 0$, $f$ and $q$ are real-valued continuous functions. Suppose that (2) is nonoscillatory at infinity and ${y_1}$, ${y_2}$ are principal, nonprincipal solutions of (2), respectively. Adapted Riccati techniques are used to obtain an asymptotic integration for the principal solution ${x_1}$ of (1). Under some mild assumptions, we characterize that (1) has a principal solution ${x_1}$ satisfying ${x_1}= {y_1}(1 + o(1))$. Sufficient (sometimes necessary) conditions under which the nonprincipal solution ${x_2}$ of (1) behaves, in three different degrees, like ${y_2}$ as $t \to \infty$ are also established.References
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Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 327 (1991), 853-865
- MSC: Primary 34E10; Secondary 34C99
- DOI: https://doi.org/10.1090/S0002-9947-1991-1028756-7
- MathSciNet review: 1028756