For functions $f(x)$ of $x$ and $g(y)$ of $y$, where $x,y$ are independent, why cannot we conclude from the expression $$ f(x)dx+g(y)dy=0\tag{1}$$ that $$f(x)=g(y)=0$$
If the original equation was $$ f(x)+g(y)=0\tag{2}$$ then we can say each function must be zero separately, otherwise a change in $x$ for a constant $y$ cannot always lead to (2)
But why is this not true for (1)?