\[ -e^{-x^2} x+y'(x)+2 x y(x)=0 \] ✓ Mathematica : cpu = 0.0133891 (sec), leaf count = 30
\[\left \{\left \{y(x)\to \frac {1}{2} e^{-x^2} x^2+c_1 e^{-x^2}\right \}\right \}\] ✓ Maple : cpu = 0.005 (sec), leaf count = 18
\[ \left \{ y \left ( x \right ) = \left ( {\frac {{x}^{2}}{2}}+{\it \_C1} \right ) {{\rm e}^{-{x}^{2}}} \right \} \]
\begin {equation} \frac {dy}{dx}+2xy\left ( x\right ) =e^{-x^{2}}x \tag {1} \end {equation}
Integrating factor \(\mu =e^{\int 2xdx}=e^{x^{2}}\). Hence (1) becomes
\begin {align*} \frac {d}{dx}\left ( e^{x^{2}}y\left ( x\right ) \right ) & =e^{x^{2}}e^{-x^{2}}x\\ \frac {d}{dx}\left ( e^{x^{2}}y\left ( x\right ) \right ) & =x \end {align*}
Integrating both sides
\begin {align*} e^{x^{2}}y\left ( x\right ) & =\frac {x^{2}}{2}+C\\ y\left ( x\right ) & =e^{-x^{2}}\left ( \frac {x^{2}}{2}+C\right ) \end {align*}