Solve $\frac{dy}{dx} - \frac{dx}{dy} = \frac{y}{x} - \frac{x}{y}$ Solve:
$$\frac{dy}{dx} - \frac{dx}{dy} = \frac{y}{x} - \frac{x}{y}$$

What I have done till now:
$$\left(\frac{dy}{dx}\right)^2  -1= \frac{dy}{dx}\left(\frac{y}{x} - \frac{x}{y}\right)$$
$$xy\left(\frac{dy}{dx}\right)^2  - xy= \frac{dy}{dx}\left({y^2} - {x^2}\right)$$
Now I am stuck.

EDIT: I have been through a course on ODEs (More like a course which added tools to our toolbox to solve ODEs) but it has been a long time since then. Everytime someone helps me with an answer it is an "Oh Yes! I forgot about that" moment. Is there any resource (Succinct PDF preferably) which can help me revise the tools used for solving ODEs? This looked good but I want more examples and methods.
A hint: If $u$ and $v$ are real numbers $\ne0$ then
$$v-{1\over v}=u-{1\over u}$$
holds iff either $v=u$ or $v=-{\displaystyle{1\over u}}$.
The differential equation reads

$$f\left(\frac{\partial y}{\partial x}\right)=f\left(\frac{y}{x}\right)$$
and 
$$\frac{\partial y}{\partial x}=\frac{y}{x}$$
has solution
$$y(x)=c\ x.$$

The kernel of the right hand side of the equation is a factor of $\pm 1$, which gets sucked up by $c$ already.