Node 2:

you can substitute implicitly (e.g. $y D_x y + (D_x^2 y)^2=0$ sub $v=y D_x y \Rightarrow D_x v = (D_x y) ^2 + y D_x^2 y \Rightarrow D_x v = 0 \Rightarrow y D_x y = y \frac{dy}{dx}=v=c \Rightarrow \int ydy= \int cdx + C$)