TITLE: What is the meaning of $\;x\;dx=y\;dy\;$?
QUESTION [5 upvotes]: I know that $x > y$ and $\;x\;dx=y\;dy.\;$ 
Can someone explain to me what is the meaning of this and how are x and y related?

REPLY [4 votes]: Given $x > y$ and $x \;dx = y\;dy$, 
First, we can integrate both sides of the equation to get $$\int x\,dx = \int y\,dy \iff \dfrac {x^2}{2} + C_1= \dfrac{y^2}{2} + C_2 \quad C_1, C_2 \text{ constants}$$ $$\implies x^2 - y^2 = 2(C_2 - C_1) = k$$
Clearly, $k > 0$ since $x> y$.
For any given $k$, the set of ordered pairs $(x, y)$ that satisfy $x^2 - y^2 = k$ is an hyperbola.