General solution of the differential equation $\frac{dy}{dx}=e^{\frac{x^2}{2}}+xy$ is

$y = (x+c)e^{\frac{x^2}{2}}$, where c is constant of integration.

The correct answer is Option (3) → $y = (x+c)e^{\frac{x^2}{2}}$, where c is constant of integration.

$\frac{dy}{dx} = e^{\frac{x^{2}}{2}} + xy$

$\text{Linear DE: } \frac{dy}{dx} - xy = e^{\frac{x^{2}}{2}}$

$\text{Integrating factor: } IF = e^{\int -x\,dx} = e^{-\frac{x^{2}}{2}}$

$\text{Multiply both sides: } e^{-\frac{x^{2}}{2}}\frac{dy}{dx} - xe^{-\frac{x^{2}}{2}}y = 1$

$\text{LHS becomes } \frac{d}{dx}(ye^{-\frac{x^{2}}{2}})$

$\frac{d}{dx}(ye^{-\frac{x^{2}}{2}}) = 1$

$ye^{-\frac{x^{2}}{2}} = x + c$

$y = (x+c)e^{\frac{x^{2}}{2}}$