Math question! You can check, for example, that y(x) + x y'(x) = ( x y(x) )' and
(x (2 y(x) - x y'(x)))/y(x)^2=(x^2/y(x))'.

But how do you find a right hand side, given a left? Are there standard methods? Say I give you a rational function F, and ask you to find a G such that

F(x,y(x),y'(x)) = d/dx G(x,y(x))

for all smooth enough y. How might you go about that? Any references? Thanks /sci !