Answer
$${\text{No linear}}{\text{, No separable}}$$
Work Step by Step
$$\eqalign{
& x\frac{{dy}}{{dx}} + y = {e^x}\left( {1 + y} \right) \cr
& {\text{divide both sides by }}x \cr
& \frac{{dy}}{{dx}} + \frac{1}{x}y = {e^x}\left( {1 + y} \right) \cr
& {\text{The equation cannot be written in the form }}\frac{{dy}}{{dx}} + P\left( x \right)y = Q\left( x \right)\,\,\,\left( {{\text{linear form}}} \right), \cr
& {\text{ then}}{\text{, the given equation is not linear}} \cr
& \cr
& and \cr
& x\frac{{dy}}{{dx}} + y = {e^x}\left( {1 + y} \right) \cr
& {\text{subtract }}y \cr
& x\frac{{dy}}{{dx}} = {e^x}\left( {1 + y} \right) - y \cr
& x\frac{{dy}}{{dx}} = \frac{{{e^x}\left( {1 + y} \right) - y}}{x} \cr
& \cr
& {\text{The equation can be written in the form }}\frac{{dy}}{{dx}} = \frac{{p\left( x \right)}}{{q\left( y \right)}},{\text{ then}} \cr
& {\text{the given equation is not separable}} \cr
& \cr
& {\text{No linear}}{\text{, No separable}} \cr} $$