An example in my Differential Equations textbook shows how to solve the homogenous differential equation $$ (x^2+y^2)\,dx +(x^2-xy)\,dy=0 $$
by substituting $y$ with $ux$, which I am trying to understand. The book explains that the reason we do this is so that $dy$ will equal $u\,dx + x\,du$.
The answer says that after substitution, the equation becomes $$(x-ux)\,dx + x(u\,dx + x\,du) = 0 $$ and then $$ dx + x\,du = 0$$
My question is, how did it get to $dx+x\,du =0$? Is it a typo or am I missing something?
I think it should be $ x\,dx + x\,du$ and then $dx + du$ and then $x + u$ and eventually $x + y/x$.
However, the textbook says the answer is $x\ln(x)+y=cx$.
What am I missing?