Answer
$\dfrac{e^y}{2}(\sin y-\cos y)=(x-1)e^x+c$
Work Step by Step
Need to separate the variables to determine the differential equation and then integrate the obtained equation on the both sides.
$\int \dfrac{e^{y} dy}{\csc y}=\int (xe^x) dx$
This implies that
$\int (e^y) (\sin y) dy=\int (xe^x) dx$
or, $(x-1)(e^x)+c=(\dfrac{e^y}{2})(\sin y-\cos y)$
so, $(\dfrac{e^y}{2})(\sin y-\cos y)=(x-1)e^x+c$
