\[ y'(x)=-x \left (-F\left (y(x)-\frac {1}{2} e^{-x^2} x^2\right )+e^{-x^2} x^2-e^{-x^2}\right ) \] ✓ Mathematica : cpu = 84.6118 (sec), leaf count = 180
\[\text {Solve}\left [c_1=\int _1^{y(x)} \left (-\int _1^x \frac {e^{-K[1]^2} K[1] \left (K[1]^2-1\right ) F'\left (K[2]-\frac {1}{2} e^{-K[1]^2} K[1]^2\right )}{F\left (K[2]-\frac {1}{2} e^{-K[1]^2} K[1]^2\right )^2} \, dK[1]-\frac {1}{F\left (K[2]-\frac {1}{2} e^{-x^2} x^2\right )}\right ) \, dK[2]+\int _1^x \frac {e^{-K[1]^2} K[1] \left (e^{K[1]^2} F\left (y(x)-\frac {1}{2} e^{-K[1]^2} K[1]^2\right )-K[1]^2+1\right )}{F\left (y(x)-\frac {1}{2} e^{-K[1]^2} K[1]^2\right )} \, dK[1],y(x)\right ]\]
✓ Maple : cpu = 0.944 (sec), leaf count = 34
\[ \left \{ y \left ( x \right ) ={\frac {{x}^{2}{{\rm e}^{-{x}^{2}}}}{2}}+{\it RootOf} \left ( {x}^{2}-2\,\int ^{{\it \_Z}}\! \left ( F \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}+2\,{\it \_C1} \right ) \right \} \]