On solutions of \(x''= t^{\alpha\lambda-2} x^{1+\alpha}\) starting at some positive \(t\) (Q1423456)

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.