Q. An integrating factor of the differential equation $x + \frac{dy}{dx} + y\log{x = xe^{x}x^{- \frac{1}{2}\log x},\ (x,0)}$ is
A) $x^{\log x}$
B) $\left( \sqrt{x} \right)^{\log x}$
C) $\left( \sqrt{e} \right)^{{{(log}{x)}}^{2}}$
D) $e^{x^{2}}$
Explanation
(c)
Given differential equation is
$x\frac{dy}{dx} + y\log{x = xe^{x}x^{- \frac{1}{2}\log x}}$
$\Rightarrow \ \frac{dy}{dx} + \frac{y}{x}\log{x = e^{x}}x^{- \frac{1}{2}\log x}$
Here, $P = \frac{1}{x}\log x$ and $Q = e^{x}x^{- \frac{1}{2}\log x}$
$\therefore IF = e^{\int_{}^{}{\frac{\log x}{x}dx}} = e^{\frac{({\log{x)}}^{2}}{2}} = \left( \ \sqrt{e} \right)^{({\log{x)}}^{2}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.