Hokkaido Mathematical Journal

TSUKAMOTO Ichiro,

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

Hokkaido Mathematical Journal, 38 (2009) pp.153-175

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Abstract

Given an initial condition $x(T)=A$, $x'(T)=B$ ($'=d/dt$, $0B_{\ast}$ 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 $t=0$.

MSC(Primary)34A12
MSC(Secondary)34A34
Uncontrolled KeywordsEmden and Fowler type differential equations, asymptotic behavior, an initial condition, a first order rational differential equation, a two dimensional autonomous system.