Separation of Variables

Separation of variables is a technique used to rearrange a first order differential equation into a form that can be integrated.
\[\frac{y}{x+2} \frac{dy}{dx}=e^y \]
Every factor containing  
  is moved to the side contain  
  and all the occurrences of  
  are moved to the other side. Multiply both sides by  
  and divide both sides by  
  to obtain  
. Now integrate.
\[\int ye^{-y}dy= \int (x+2)dx\]

The left hand side is integrated using the Integration By Parts method.
\[-ye^{-y}-e^{-y}= \frac{x^2}{2}+2x+c\]