On solutions of \(x''= t^{\alpha\lambda-2} x^{1+\alpha}\) starting at some positive \(t\)
type: Item
instance of: scholarly article
title: On solutions of \(x''= t^{\alpha\lambda-2} x^{1+\alpha}\) starting at some positive \(t\)
zbMATH Open document ID: 1061.34006
author: Ichiro Tsukamoto
published in: Hokkaido Mathematical Journal
publication date: 2004-03-01T00:00:00Z
review text: The author published several articles on second-order differential equations of the type \(x''(t)=t^rx^s\), where \(r\) and \(s\) are fractions, following earlier work of T. Saito in the late 1970's. Here, he considers \[ x''(t)=t^\beta x^{1+ \alpha},\quad t\in[0,\infty),\;x\in[0,\infty), \tag{1} \] noting that the famous Thomas-Fermi equation is of this form when \(\beta=-1/2\), and \(\alpha=1/2\). T. Saito established that bounded solutions with a bounded first derivative exist if and only if \(\beta>-1\). Analyticity was shown in the vicinity of \(t=0\) and \(t= \infty\). The author continued these investigations between 1980's and present time, publishing several papers on closely related problems. Following Saito, the author considers equation (1) with the initial conditions \(x(T)=A\), \(x'(T)= \beta\), with \(0<T<\infty\). Let \(\varphi(t)\) be any solution of (1) and \(\psi (t)\) be a particular solution of (1) of the form \(\psi(t)=\{\lambda (\lambda+ 1)\}^{1/ \alpha}t^{-\lambda}\). Consider the transformation \(y=\psi(t)^{-\alpha} \varphi(t)^\alpha\), \(z=ty'\). This transforms (1) into the rational differential equation \[ dz/dy=\chi(z,y)/\alpha yz, \] where \(\chi(z,y)\) is a cubic polynomial in the variables \(y\) and \(z\). Solving the equation, the author translates his results back into the original form (1). A lengthy study of the trajectories results in statements concerning the behavior of solutions in the neighborhoods of the end points of the domain. Theorems proved here have rather lengthy hypotheses, as different initial conditions are considered.
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34A25
Mathematics Subject Classification ID: 34D05
zbMATH DE Number: 2048550
zbMATH Keywords: nonlinear second-order ordinary differential equation
zbMATH Keywords: Saito transformations
zbMATH Keywords: asymptotic behavior
reviewed by: Vadim Komkov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1350659155
OpenAlex ID: W2041933977
DOI: 10.14492/HOKMJ/1350659155
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description: scientific article; zbMATH DE number 2048550