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Question
If sin x is an integrating factor of the differential equation \[\frac{dy}{dx} + Py = Q\], then write the value of P.
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Solution
\[\text{ It is given that }\sin x \text{ is the integrating factor of the differential equation }\frac{dy}{dx} + Py = Q . \]
\[ \therefore e^{\int P\ dx} = \sin x\]
\[ \Rightarrow \int P\ dx = \log \left| \sin x \right|\]
\[ \Rightarrow \int P dx = \int\cot x dx .........\left[ \because \int\cot x dx = \log \left| \sin x \right| + C \right]\]
\[ \Rightarrow P = \cot x \]
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