Solution of the differential equation $y=x \cdot \frac{dy}{dx}+\frac{b}{\frac{dy}{dx}}$ Solve the following differential equation:
$$y=x \cdot \frac{dy}{dx}+\frac{b}{\frac{dy}{dx}}$$ where $b$ is a real constant.
I tried solving it by taking $\frac{dy}{dx}=t$, then solving quadratic equation for roots. But this method gives complicated differential equation once I write expression for roots. Could someone suggest a better approach.

i don't know if this helps. 
let $m = \frac{dy}{dx}.$ now we can write your differential equation  $y=x \cdot \frac{dy}{dx}+\frac{b}{\frac{dy}{dx}}$ as $$y = xm + \frac b m  $$
solving this quadratic equation, you have $$2x\ m = y \pm \sqrt{y^2 - 4bx}\to 2x\frac{dy}{dx} = y\pm\sqrt{y^2 - 4bx}  $$ 
