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Q.
If $y=\left(\frac{2}{\pi} x-1\right) cosec x$ is the solution of the differential equation,
$\frac{ dy }{ dx }+ p ( x ) y =\frac{2}{\pi} \operatorname{cosec} x , 0< x <\frac{\pi}{2},$ then the function $p ( x )$ is equal to
$y =\left(\frac{2 x }{\pi}-1\right)cosecx\,\dots(1)$
$\frac{d y}{d x}=\frac{2}{\pi}cosec x-\left(\frac{2 x}{\pi}-1\right) cosec\, x \,cot\, x$
$\frac{d y}{d x}=\frac{2 cosec x}{\pi}-y \,cot \,x$
using equation _____ (1)
$\frac{ dy }{ dx }+ y \cot x =\frac{2 \operatorname{cosec} x }{\pi}$
$\frac{ d y }{ d x }+ p ( x ) \cdot y =\frac{2 \operatorname{cosec} x }{\pi} \quad x \in\left(0, \frac{\pi}{2}\right)$
Compare : $p ( x )=\cot \,x$