If $y(\pi)=\pi+2$, then the value of $y\left(\frac{\pi}{2}\right)$ is:
$\frac{\mathrm{dy}}{\mathrm{dx}}=|\mathrm{A}|$
$\frac{\mathrm{d} y}{\mathrm{~d} \mathrm{x}}=\frac{\mathrm{y}}{\mathrm{x}}+2 \sin \mathrm{x}+2$
$\frac{\mathrm{d} y}{\mathrm{~d} \mathrm{x}}+\frac{-\mathrm{y}}{\mathrm{x}}=2 \sin x+2$
$\text { I.F. }=\mathrm{e}^{\int \frac{1}{\mathrm{x}} \mathrm{dx}}=\mathrm{x}$
$\Rightarrow \mathrm{yx}=\int \mathrm{x}(2 \sin \mathrm{x}+2) \mathrm{d} \mathrm{x}$
$\mathrm{xy}=\mathrm{x}^{2}-2 \mathrm{x} \cos x+2 \sin x+\mathrm{c}.....(i)$
Now $x=\pi, y=\pi+2$
Use in $(i)$
$c=0$
Now $(i)$ be comes
$x y=x^{2}-2 x \cos x+2 \sin x$
put $x=\frac{\pi}{2}$
$\frac{\pi}{2} y=\frac{\pi^{2}}{4}+2$
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