Question

$\frac{dy}{dx}-y\cot x=\mathrm{cosec}x$

Answer 1

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