Question:
The integrating factor of the differential equation $x \frac{d y}{d x}-y=2 x^{2}$ is
A. $e^{-x}$
B. $e^{-y}$
C. $\frac{1}{x}$
D. $x$
Solution:
The given differential equation is:
$x \frac{d y}{d x}-y=2 x^{2}$
$\Rightarrow \frac{d y}{d x}-\frac{y}{x}=2 x$
This is a linear differential equation of the form:
$\frac{d y}{d x}+p y=Q\left(\right.$ where $p=-\frac{1}{x}$ and $\left.Q=2 x\right)$
$e^{\int p d x}$
$\therefore$ I.F $=e^{\int-\frac{1}{x} d x}=e^{-\log x}=e^{\log \left(x^{-1}\right)}=x^{-1}=\frac{1}{x}$
Hence, the correct answer is C.