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Asymptotic behavior of solutions of \(x''=e^{\alpha\lambda t}x^{1+\alpha}\) where \(-1<{\alpha}<0\). (English) Zbl 1081.34044

The paper is devoted to ordinary differential equation \[ x''=e^{\alpha\lambda t}x^{1+\alpha}. \] The main results concern the initial value problem \[ x(t_0)=a,\;x'(t_0)=b. \] The author studies the domain of definition and asymptotic expansion of solutions. His main results are formulated in two different cases given by conditions on the parameter values (Theorems I and II). Theorem I says that under appropriate conditions on the parameters, the solution is well-defined on the whole real line and in neighborhoods of \(\pm\infty\) it admits an expansion in series. The expansions at \(+\infty\) and \(-\infty\) are different in general. Moreover, in one of the cases considered in Theorem I the expansion near \(-\infty\) is given not by a series, but just by an asymptotic formula. Theorem II says that under some different conditions on the parameter values, the solution is well-defined on an interval \((\omega_-,+\infty)\), \(\omega_->-\infty\), and admits an expansion similar to the one mentioned above at each boundary point of the domain of definition (the expansion depends on the choice of the boundary point).

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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