On solutions of \(x''=-e^{\alpha \lambda t}x^{1+\alpha}\)
The nonlinear differential equations \[ \frac{d}{dt}(t^ p\frac{du}{dt})+t^ su^ n=0\quad and\quad \frac{d}{dt}(t^ p\frac{du}{dt})-t^ su^ n=0 \] are the starting point. These equations can be transformed into four differential equations. The author considers one - that means the last - case of the for cases. By the transformation \(y=-\lambda^{-2}e^{\alpha \lambda t}\phi^{\alpha},\) \(z=y'\) and \(\phi\) is an arbitrary solution of the given equation you get a first order algebraic differential equation and you can start to discuss the asymptotic behavior of solutions.
first order algebraic differential equation
scientific article; zbMATH DE number 4134512