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Question
\[\frac{dy}{dx} - y \cot x = cosec\ x\]
Solution
We have,
\[\frac{dy}{dx} - y \cot x = cosec\ x\]
\[\text{Comparing with }\frac{dy}{dx} + Py = Q,\text{ we get}\]
\[P = - \cot x \]
\[Q = cosec\ x\]
Now,
\[I . F . = e^{\int - \cot x\ dx} \]
\[ = e^{- \log \left| \left( \sin x \right) \right|} \]
\[ = e^{\log \left| \left(cosec\ x \right) \right|} \]
\[ = cosec x\]
So, the solution is given by
\[y\ cosec\ x = \int cosec\ x \times cosec\ x\ dx + C\]
\[ \Rightarrow y\ cosec\ x = \int {cosec}^2 x dx + C\]
\[ \Rightarrow y\ cosec\ x = - \cot x + C\]
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