Given differential equations,
\[
\begin{array}{l}
x \frac{d y}{d x}+2 y=x^{2} \\
\Rightarrow \frac{d y}{d x}+\frac{2 y}{x}=x \rightarrow(1)
\end{array}
\]
General differential equation of this type is given by
\[
\frac{d y}{d x}+P(x) y=Q(x)
\]
So, \( P=\frac{2}{x} \) then I.F. factor is given by
\[
e^{\int P d x}=e^{\int \frac{2}{x} d x}=e^{2 \log x}=e^{\log x^{2}}=x^{2}
\]
Then, \( x^{2} \frac{d y}{d x}+2 x y=x^{3} \)
\[
\Rightarrow \frac{d}{d x}\left(x^{2} y\right)=x^{3}
\]
Integrating both the sides, we get
\[
\begin{array}{l}
x^{2} y=\frac{x^{4}}{4}+C_{1} \\
\Rightarrow x^{2} y=\frac{x^{4}+C}{4} \\
\Rightarrow y=\frac{x^{4}+C}{4 x^{2}}
\end{array}
\]