$\cfrac{{dy}}{{dx}} + \cfrac{y}{x} = {x^2}$

: The given equation is $\cfrac{{dy}}{{dx}} + \cfrac{y}{x} = {x^2}$

…(1)
which is a linear equation of the type $\cfrac{{dy}}{{dx}} + Py = Q$

Where, $P = \cfrac{1}{x}$ and $Q = {x^2}.$

$\therefore {\rm{I}}{\rm{.F}}{\rm{.}} = {e^{\int P dx}} = {e^{\int {\cfrac{1}{x}} dx}} = {e^{\log x}} = x$

$\therefore$

The solution is $y \cdot \left( {{\rm{I}}{\rm{.F}}{\rm{.}}} \right) = \int Q \cdot \left( {{\rm{I}}{\rm{.F}}{\rm{.}}} \right)dx + C$

$\Rightarrow yx = \int {{x^3}} dx + C \Rightarrow yx = \cfrac{1}{4}{x^4} + C$,

which is the required solution.