I am trying to show this kind of non-linear $y''''=y'y''/(1+x)$ in normal form. For example here if $y=e^{x}\rightarrow y^{(n)}=e^{x}\rightarrow x=-1$, where $y^{(n)}$ means $n$th differential, then $x=-1$, too weak idea. When I google with differential or anything like that, most of the material does not look the material that I need. I need to solve different type of problems such as this homework
$$\begin{cases} u'=(u+v)^{2} \\ v''=x+u'v' \end{cases} $$
I am not requesting you to solve them but I am requesting some material because I find my book quite hard-reading in this section. The earlier chapter begun that something is something, without much further ado really why?, and now the next advanced chapters are referring to the past chapters. The idea is a rush introduction to this topic in an engineering course so I think it explains quite a bit about the pedagogy.