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