JEE PYQ: Differential Equation Question 26
Question 26 - 2020 (06 Sep Shift 2)
If $y = \left(\frac{2}{\pi}x - 1\right)\operatorname{cosec} x$ 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:
(1) $\cot x$
(2) $\operatorname{cosec} x$
(3) $\sec x$
(4) $\tan x$
Show Answer
Answer: (1)
Solution
$\frac{dy}{dx} = \frac{2}{\pi}\operatorname{cosec} x - \left(\frac{2}{\pi}x - 1\right)\operatorname{cosec} x\cot x$. Substituting into the DE: $\frac{2}{\pi}\operatorname{cosec} x - y\cot x + p(x)y = \frac{2}{\pi}\operatorname{cosec} x$. So $p(x) = \cot x$.
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