On solutions of \(x''=t^{\alpha\lambda-2}x^{1+\alpha}\) in the unsettled cases. (English) Zbl 1470.34100

Tsukamoto, Ichiro

Yokohama Math. J. 65, 77-84 (2019).

The author considers the following Emden-Fowler type differential equation \[ x''=t^{\alpha\lambda-2}x^{1+\alpha}. \tag{1} \] Here \(\alpha\) and \(\lambda\) are real parameters, \(t, x\)- are positive variables.
The article consists of 3 sections and references. The first section is devoted to a general introduction into the problem. The second section is devoted to the asymptotic representation of solutions of the equation (1) in the cases \[ \alpha<\lambda_0,\, \lambda<-1\text{ or }\alpha<\lambda_0,\, \lambda>0\ (\lambda_0=-(2\lambda+1)^2/4\lambda (\lambda+1)). \] The third section is devoted to the asymptotic representation of solutions of the equation (1) in case \[ \alpha<0,\, -1<\lambda<0. \]
The asymptotic formulas for the solutions of the equation (1) are obtained in the form of a series. Corresponding results were obtained in the works of G. Sansone [in: Equadiff 78, Conv. int. su equazioni differenziali ordinarie ed equazioni funzionali, Firenze 1978, Suppl. separati, 38 p. (1978; Zbl 0429.34003)] and V. M. Evtukhov [The asymptotic behavior of solutions of a nonlinear second order differential equation of the Emden-Fowler type. Odessa: Odessa State University (Diss.) (1980)].
In my opinion the results presented in the article are very important for the further study of asymptotic representations of solutions of n-th order differential equations of the Emden-Fowler type.

Reviewer: Olga Chepok (Odessa)

MSC:

34C11	Growth and boundedness of solutions to ordinary differential equations
34D05	Asymptotic properties of solutions to ordinary differential equations

Keywords:

asymptotic behaviour; analytical expression; initial value problem

Citations:

Zbl 0429.34003