On asymptotic behavior of positive solutions of $x''=-t^{\alpha\lambda-2}x^{1+\alpha}$ with $\alpha\lt 0$ and $\lambda=0, -1$

Abstract

Given an initial condition $x(T)=A$, $x'(T)=B$ ($'=d/dt$, $0\lt T \lt \infty$, $0 \lt A \lt \infty$, $-\infty \lt B \lt \infty$) for the differential equation denoted in the title, we shall conclude that if $T$, $A$ are fixed arbitrarily, then there exists a number $B_{\ast}$ such that in every case of $B=B_{\ast}$, $B \lt B_{\ast}$, $B \gt B_{\ast}$ we determine analytical expressions of the solution of the initial value problem which shows asymptotic behavior of the solution. That is, these analytical expressions are valid in neighborhoods of ends of the domain of the solution. If $\lambda =-1$, then we shall treat the case $T=0$, since there exists the solution continuable to $t=0$.