Question

The integrating factor of $\frac{dy}{dx}+y=\frac{1+y}{x}$ is

• A. $x{e}^x$
• B. $x{e}^{1/x}$
• C. $\frac{e^x}{x}$
• D. $\frac{x}{e^x}$

Answer 1

The correct option is C $\frac{e^x}{x}$

$\frac{dy}{dx}+y=\frac{1+y}{x}$
$\Rightarrow \frac{dy}{dx}+y=\frac{1}{x}+\frac{y}{x}$
$\Rightarrow \frac{dy}{dx}+(1-\frac{1}{x})y=\frac{1}{x}$
$I.F.=\exp (\int (1-\frac{1}{x}\left)dx\right)$
$=\exp (x-\log x)$
$=e^x.e^{-\log x}=\frac{e^x}{x}$