Asymptotic behavior of positive solutions of \(x''= t^{\alpha\lambda-2} x^{1+\alpha}\) in the sublinear case (Q989211)

The author studies the initial value problem \[ x''(t) = t^{\alpha \lambda -2}x(t)^{1+\alpha }, \quad 0<t<\infty , \; 0<x<\infty, \] \[ x(t_{0}) = A, \qquad x'(t_{0}) = B, \] where \(0<t_{0}<\infty \), \(0<A<\infty \), \(-\infty <B<\infty \) and \[ -(2\lambda +1)^{2}/4\lambda (\lambda +1)<\alpha <0 , \; \lambda <-1. \] He obtains a series representation for the solution \(x(t)\) both in the neighborhood of \(x=0\) and of \(x=\infty \). There are two different cases to be considered depending on the choice of \(A\) and \(B\). The proof consists of representing the considered equation as a 2-dimensional autonomous system and then studying its orbits around its critical points.