Question

The integrating factor of the differential equation $x\frac{dy}{dx}-y$ = sin x is ____________.

Answer 1

Given: $x\frac{dy}{dx}-y$ = sinx

$$\begin{gathered} x\frac{dy}{dx}-y=\sin x \\ \Rightarrow \frac{dy}{dx}-\frac{y}{x}=\frac{\sin x}{x} \\ \mathrm{Here}P=-\frac{1}{x}\mathrm{and}Q=\frac{\sin x}{x} \\ \mathrm{Integrating}\mathrm{Factor}=e^{\int Pdx} \\ =e^{\int -\frac{1}{x}dx} \\ =e^{-\log x} \\ =e^{\log {\left(x\right)}^{-1}} \\ =x^{-1} \\ =\frac{1}{x} \end{gathered}$$
Hence, the integrating factor of the differential equation $x\frac{dy}{dx}-y$ = sinx is $\underline{\frac{1}{x}}$.