Question

If sin x is an integrating factor of the differential equation $\frac{dy}{dx}+Py=Q$, then write the value of P.

Answer 1

$$\begin{gathered} \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}\sin x\mathrm{is}\mathrm{the}\mathrm{integrating}\mathrm{factor}\mathrm{of}\mathrm{the}\mathrm{differential}\mathrm{equation}\frac{dy}{dx}+Py=Q. \\ \therefore e^{\int Pdx}=\sin x \\ \Rightarrow \int Pdx=\log \left|\sin x\right| \\ \Rightarrow \int Pdx=\int \cot xdx\left[\because \int \cot xdx=\log \left|\sin x\right|+C\right] \\ \Rightarrow P=\cot x \end{gathered}$$