Question

\[\frac{dy}{dx} - y \cot x = cosec\ x\]

Answer 1

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

\[y\ cosec\ x = \int cosec\ x \times cosec\ x\ dx + C\]

\[ \Rightarrow y\ cosec\ x = \int {cosec}^2 x dx + C\]

\[ \Rightarrow y\ cosec\ x = - \cot x + C\]