## Separation of Variables

\[\frac{y}{x+2} \frac{dy}{dx}=e^y \]

.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.

\[-ye^{-y}-e^{-y}= \frac{x^2}{2}+2x+c\]