The integrating factor of the differential equation $\frac{dy}{dx}=x+xy$ is

$e^{-x^2/2}$

The correct answer is Option (2) → $e^{-x^2/2}$

Given differential equation:

$\frac{dy}{dx}=x+xy$

Rewrite in linear form:

$\frac{dy}{dx}-xy=x$

This is of the form $\frac{dy}{dx}+P(x)y=Q(x)$ with:

$P(x)=-x$

Integrating factor (I.F.) = $e^{\int P(x)\,dx}$

I.F. = $e^{\int -x\,dx} = e^{-\frac{x^{2}}{2}}$

The integrating factor is $e^{-\frac{x^{2}}{2}}$.