Hey anyone can help me with this problem? My friend asked me to find the implicit function of this:
$$(x+2y)\,dx - 2xy\,dy = 0 \qquad\text{(1)}$$
My approach is to use a function $u(x,y)$ then $du = u_x \, dx + u_y \, dy$. Then with (1) I have $u_x = x+2y$ and $u_y = -2xy$ (2). But then I can't find the function u which satisfied the condition (2). So I wonder are there any conditions for the implicit function? And how should I deal with (1)?
The differential form $$(x+2y) dx +2xy dy$$ is not closed, hence the differential form cannot be exact. In fact $$\frac{\partial}{\partial y} (x+2y)=2$$ $$\frac{\partial}{\partial x}(2xy)=2y$$ More explicitly, if a potential function $u$ existed then $$u_x=x+2y \implies u=\frac 12 x^2+2xy+C_1(y)$$ and $$u_y=2xy \implies u=xy^2+C_2(x)$$ The two conditions are clearly not compatible.