If tan x is an integrating factor of the diff | vedclass

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(D) The integrating factor $(IF)$ of the linear differential equation $\frac{dy}{dx} + Py = Q$ is given by $IF = e^{\int P \, dx}$.Given $IF = 	an x$

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$2 \csc 2x$

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(D) The integrating factor $(IF)$ of the linear differential equation $\frac{dy}{dx} + Py = Q$ is given by $IF = e^{\int P \, dx}$.Given $IF = 	an x$, we have $e^{\int P \, dx} = 	an x$.Taking the natural logarithm on both sides: $\int P \, dx = \ln(	an x)$.Differentiating both sides with respect to $x$: $P = \frac{d}{dx}(\ln(	an x))$.Using the chain rule: $P = \frac{1}{	an x} \cdot \sec^2 x$.$P = \frac{\cos x}{\sin x} \cdot \frac{1}{\cos^2 x} = \frac{1}{\sin x \cos x}$.Multiply numerator and denominator by $2$: $P = \frac{2}{2 \sin x \cos x} = \frac{2}{\sin 2x} = 2 \csc 2x$.

If $\tan x$ is an integrating factor of the differential equation $\frac{dy}{dx} + Py = Q$, then $P$ can be

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