On a modification of the small parameter method (Q2461862)

There is an extensive class of the differential equations, where a classical method to study analytical properties of their solutions -- the method of small parameter -- is not effective. For example, the equations of so-called barrier type \[ \begin{aligned} w'''&=ww''-2w^{\prime 2},\\ w^{(4)}&=w'''w-3w''w'+c(w^{\prime 2} w-2w''w), \quad c\in\mathbb{C},\\ w^{(4)}&=w'''w^2-7w''w'w+6w^{\prime 3},\end{aligned} \] belong to this class. In such cases the author suggests to use the method of small parameter multiplying the highest derivative. He proves by means of this method that the solutions of all barrier equations admit critical movable points.