Asymptotic Behavior of Positive Solutions of $x''=t^{\alpha\lambda-2}x^{1+\alpha}$ in the Sublinear Case

Abstract

In this paper, we shall consider an initial value problem of the differential equation written in the title where \[ '=d/dt ,\quad -(2\lambda+1)^2/4\lambda(\lambda+1) < \alpha < 0 ,\quad \lambda < -1. \] For every initial condition, we shall obtain analytical expressions of the solution of the initial value problem. These analytical expressions are valid in neighborhoods of both ends of the domain of the solution, and hence show asymptotic behavior of the solution.