Looking for an intuitive approach of ODE I started reading up on the topics of differential equations and tried to solve certain problems to get used to those kind of equations, in particular I tried to understand every "$=$" and "$\Rightarrow$". 
However I soon stumbled over this equation:
$ {1 \over y} * { dx \over dy } - {x \over y^2}   = y $
or: $ {d \over dy} ( {x \over y} ) = y $
Now I am trying to understand why: $ {1 \over y} * { dx \over dy } - {x \over y^2} = {d \over dy} ( {x \over y} ) $
In particular I am confused how to interpret those two terms, how to interpret them: 
${d \over dy} ( {x \over y} ) $ and $ { dx \over dy } $ ; 
I would be happy, if someone could help me and might recommend a detailed read about differential equation.  

$x$ depends on $y$.
So :
$$\frac{d}{dy} \left( \frac{x}{y} \right)=\frac{d}{dy} \left( x \cdot \frac{1}{y} \right)=\frac{dx}{dy} \cdot \frac{1}{y}+x \frac{d}{dy}\left(\frac{1}{y} \right)=\frac{dx}{dy} \cdot \frac{1}{y}+x \left(-\frac{1}{y^2} \right)$$
