Solve this following


If $y=\left(\frac{2}{\pi} x-1\right) \operatorname{cosecx}$ is the solution of the

differential equation,

$\frac{d y}{d x}+p(x) y=\frac{2}{\pi} \operatorname{cosec} x, 0

function $\mathrm{p}(\mathrm{x})$ is equal to


  1. $\cot x$

  2. $\tan x$

  3. $\operatorname{cosecx}$

  4. $\operatorname{secx}$

Correct Option: 1,


$y=\left(\frac{2 x}{\pi}-1\right) \operatorname{cosecx}$ $\ldots(1)$

$\frac{d y}{d x}=\frac{2}{\pi} \operatorname{cosecx}-\left(\frac{2 x}{\pi}-1\right) \operatorname{cosecx} \cot x$

$\frac{d y}{d x}=\frac{2 \operatorname{cosecx}}{\pi}-y \cot x$

using equation (1)

$\frac{d y}{d x}+y \cot x=\frac{2 \operatorname{cosec} x}{\pi}$

$\frac{d y}{d x}+p(x) \cdot y=\frac{2 \operatorname{cosec} x}{\pi} \quad x \in\left(0, \frac{\pi}{2}\right)$

Compare : $p(x)=\cot x$

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