On the theory of Painlevé’s second equation. (English. Russian original) Zbl 0554.34032

Abdullaev, A. S.

Sov. Math., Dokl. 28, 726-729 (1983); translation from Dokl. Akad. Nauk SSSR 273, 1033-1036 (1983).

The author proves that if \(| d| <3^{1/2}\), then the asymptotic formula \(w(x)=dx^{-1/4}\sin \phi_ 0(x)+O(x^{-1/4})\), \(\phi_ 0(x)=(2/3)x^{3/2}-(3/4)d^ 2\ln (x)+c_ 0+O(x^{-3/2})\) holds for a solution of \(w''+xw=2w^ 3\) as \(x\to \infty\), from which the author gives a more general asymptotic formula for some classes of second order nonlinear equations.

Reviewer: T.S.Liu

MSC:

34E05

Asymptotic expansions of solutions to ordinary differential equations

Keywords:

asymptotic formula; second order nonlinear equations