#### Answer

$${\text{Separable}}$$

#### Work Step by Step

$$\eqalign{
& \frac{x}{y}\frac{{dy}}{{dx}} = 4 + {x^{3/2}} \cr
& {\text{multiply both sides by }}\frac{y}{x} \cr
& \frac{{dy}}{{dx}} = \frac{{y\left( {4 + {x^{3/2}}} \right)}}{x} \cr
& or \cr
& \frac{{dy}}{{dx}} = \frac{{{x^{ - 1}}\left( {4 + {x^{3/2}}} \right)}}{y} \cr
& \cr
& {\text{The equation can be written in the form }}\frac{{dy}}{{dx}} = \frac{{p\left( x \right)}}{{q\left( y \right)}},{\text{ then}} \cr
& {\text{the given equation is separable}} \cr
& \cr
& {\text{The equation cannot be written in the form }}\frac{{dy}}{{dx}} + P\left( x \right)y = Q\left( x \right)\,\,\,\left( {{\text{linear form}}} \right), \cr
& {\text{ then}}{\text{, the given equation is not linear}} \cr
& \cr
& {\text{Separable}} \cr} $$