Question

The integrating factor of the differential equation $\frac{dy}{dx}\left(x{\log }_{e}x\right)+y=2{\log }_{e}x$ is given by

• A. $x$
• B. $e^x$
• C. $|{\log }_{e}x|$
• D. ${\log }_e|{\log }_{e}x|$

Answer 1

The correct option is C $|{\log }_{e}x|$

$\because \frac{dy}{dx}+\frac{y}{x{\log }_{e}x}=\frac{2}{x}$
$\therefore I.F.=e^{\int \frac{1}{x{\log }_{e}x}dx}$
$=e^{{\log }_e|{\log }_{e}x|}=|{\log }_{e}x|$