If y(x) satisfy the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{\mathrm{y}}{\mathrm{x}}=\cos \mathrm{x}+\frac{\sin \mathrm{x}}{\mathrm{x}}$ and y(0)=0 then

$\begin{array}{l}\frac{dy}{dx}+\left(\frac{1}{x}\right)y=\cos x+\frac{\sin x}{x}\\ I.F=e^{\int \frac{1}{x}dx}=e^{\log x}=x\\ Solutionis\\ y(I.F)=\int (\cos x+\frac{\sin x}{x})(I.F)dx\\ \Rightarrow xy=\int (\cos x+\frac{\sin x}{x})xdx\\ =\int (x\cos x+\sin x)dx\\ =x\sin x-\int 1.\sin xdx-\cos x+c\\ =x\sin x+\cos x-\cos x+c\\ \Rightarrow y.x=x\sin x+c\end{array}$

$\begin{array}{l}Putx=0\\ y\left(0\right)\left(0\right)=0\sin x+c\\ \Rightarrow c=0\\ \therefore y.x=x\sin x\\ y=\sin x\\ y\left(\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)=\sin \left(\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)=1\\ y\left(\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right)=\sin \left(\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right)=\frac{1}{\sqrt{2}}\end{array}$