$\text{ The solution of the differential equation }\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}+2\mathrm{y}={\mathrm{x}}^2,(\mathrm{x}\ne 0)\text{ with }\mathrm{y}\left(1\right)=1\text{ then y(2) is}$

$\begin{array}{l}\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{2}{\mathrm{x}}\mathrm{y}=\mathrm{x}\\ \text{ I. }F=e^{\int \frac{2}{x}dx}=e^{2\mathrm{logx}}=x^2\\ \text{ Solution of the D.E in }\mathrm{y}.{\mathrm{x}}^2=\int \mathrm{x}\cdot {\mathrm{x}}^2\mathrm{dx}\\ x^{2}y=\frac{x^4}{4}+c\\ y=\frac{x^2}{4}+\frac{c}{x^2}\\ \text{ If y}\left(1\right)=1\Rightarrow 1=\frac{1}{4}+c\to c=\frac{3}{4}\\ \mathrm{y}\left(\mathrm{x}\right)=\frac{{\mathrm{x}}^2}{4}+\frac{3}{4{\mathrm{x}}^2}\Rightarrow \mathrm{y}\left(2\right)=\frac{4}{4}+\frac{3}{16}\end{array}$

$\text{=}\frac{\text{16+3}}{16}\text{=}\frac{19}{16}\text{=1.11}$