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

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:

 3.4e+06 Asymptotic expansions of solutions to ordinary differential equations

### Keywords:

asymptotic formula; second order nonlinear equations