\[ -e^{-x^2} x+y'(x)+2 x y(x)=0 \] ✓ Mathematica : cpu = 0.0389584 (sec), leaf count = 30
DSolve[-(x/E^x^2) + 2*x*y[x] + Derivative[1][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {1}{2} e^{-x^2} x^2+c_1 e^{-x^2}\right \}\right \}\] ✓ Maple : cpu = 0.007 (sec), leaf count = 18
dsolve(diff(y(x),x)+2*x*y(x)-x*exp(-x^2) = 0,y(x))
\[y \left (x \right ) = \left (\frac {x^{2}}{2}+c_{1}\right ) {\mathrm e}^{-x^{2}}\]
Hand solution
\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*}