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  
\[x\]
  is moved to the side contain  
\[dy\]
  and all the occurrences of  
\[x\]
  are moved to the other side. Multiply both sides by  
\[x+2\]
  and divide both sides by  
\[e^y\]
  to obtain  
\[ye^{-y}dy=(x+2)dx\]
. Now integrate.
\[\int ye^{-y}dy= \int (x+2)dx\]

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