Question

The Integrating Factor (IF) of the differential equation $x\frac{dy}{dx}-y=2x^2$ is
(a)$e^{-x}$
(b)$e^{-y}$
(c)$\frac{1}{x}$
(d)x

Answer 1

Given, differential equation is $x\frac{dy}{dx}-y=2x^2\Rightarrow \frac{dy}{dx}+(-\frac{1}{x})y=2x$
It is a linear differential equation of the form $\frac{dy}{dx}+Py=Q$
$\therefore P=-\frac{1}{x}$ and Q=2x and $IF=e^{\int Pdx}\Rightarrow IF=e^{\int (-\frac{1}{x})dx}\Rightarrow IF=e^{-log|x|}\Rightarrow IF=e^{log|x{|}^{-1}}=\frac{1}{x}$
An integrating factor for the given equation is $\frac{1}{x}$(assuming x>0).
Hence, the correct option is (c).