\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac { \left ( -8\,{{\rm e}^{-{x}^{2}}}y \left ( x \right ) +4\,{x}^{2} \left ( {{\rm e}^{-{x}^{2}}} \right ) ^{2}-8\,{{\rm e}^{-{x}^{2}}}+8\,{x}^{2}{{\rm e}^{-{x}^{2}}}y \left ( x \right ) -4\,{x}^{4} \left ( {{\rm e}^{-{x}^{2}}} \right ) ^{2}+8\,{x}^{2}{{\rm e}^{-{x}^{2}}}-8\, \left ( y \left ( x \right ) \right ) ^{3}+12\,{x}^{2}{{\rm e}^{-{x}^{2}}} \left ( y \left ( x \right ) \right ) ^{2}-6\,y \left ( x \right ) {x}^{4} \left ( {{\rm e}^{-{x}^{2}}} \right ) ^{2}+{x}^{6} \left ( {{\rm e}^{-{x}^{2}}} \right ) ^{3} \right ) x}{-8\,y \left ( x \right ) +4\,{x}^{2}{{\rm e}^{-{x}^{2}}}-8}}=0} \]
Mathematica: cpu = 0.081510 (sec), leaf count = 150 \[ \left \{\left \{y(x)\to \frac {e^{-3 x^2}}{8 \left (\frac {1}{8} e^{-3 x^2}-\frac {e^{-3 x^2}}{\sqrt {c_1-64 x^2}}\right )}-\frac {1}{2} e^{-x^2} \left (2 e^{x^2}-x^2\right )\right \},\left \{y(x)\to \frac {e^{-3 x^2}}{8 \left (\frac {e^{-3 x^2}}{\sqrt {c_1-64 x^2}}+\frac {1}{8} e^{-3 x^2}\right )}-\frac {1}{2} e^{-x^2} \left (2 e^{x^2}-x^2\right )\right \}\right \} \]
Maple: cpu = 0.125 (sec), leaf count = 100 \[ \left \{ y \left ( x \right ) ={\frac {1}{2} \left ( {{\rm e}^{-{x}^{2}}} \sqrt {-{x}^{2}+{\it \_C1}}{x}^{2}-{x}^{2}{{\rm e}^{-{x}^{2}}}+2 \right ) \left ( -1+\sqrt {-{x}^{2}+{\it \_C1}} \right ) ^{-1}},y \left ( x \right ) ={\frac {1}{2} \left ( {{\rm e}^{-{x}^{2}}}\sqrt {-{x }^{2}+{\it \_C1}}{x}^{2}+{x}^{2}{{\rm e}^{-{x}^{2}}}-2 \right ) \left ( 1+\sqrt {-{x}^{2}+{\it \_C1}} \right ) ^{-1}} \right \} \]