I am puzzled by this. You are asking about the differential equation \(\displaystyle M(x,y)dx+ N(x,y)dy= 0\) but appear to be saying that you do not know what "dx" and "dy" mean!
The "dx" and "dy" here are differentials and satisfying \(\displaystyle dy= f'(x)dx\). That is the same as \(\displaystyle \frac{dy}{dx}= f'(x)\). Given the equation \(\displaystyle M(x,y)dx+ N(x,y)dy= 0\), divide through by dx to get \(\displaystyle M(x,y)+ N(x,y)\frac{dy}{dx}= M(x,y)+ N(x,y)y'= 0\).