In this case $u=u(x,y)$. When I saw this I just went on to taking iindefinite integral both sides yielding $ u^2=4xy+K $. Yet, the book I am using now got $udu=d(xy)$, which yields $ u^2=2xy+K$. I'm I right or the book is?
These are the details of what I did. $ \displaystyle \int udu = \int ydx +xdy $ which implies that $ \displaystyle \frac{u^2}{2}= xy +xy + K$
However, this is what I think they did to get the RHS.
RHS= $ \langle y, x \rangle \cdot d \langle x, y \rangle = \nabla (xy) \cdot d (\overrightarrow { r}) =d(xy)$.
However, I think my reasoning is okay? Help please.
The screen copy below shows where is the mistake :