\[ \boxed { {\it d4y} \left ( x \right ) +4\,ax{\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}y \left ( x \right ) +6\,{a}^{2}{x}^{2}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +4\,{a}^{3}{x}^{3}{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +{a}^{4}{x}^{4}y \left ( x \right ) =0} \]
Mathematica: cpu = 0.649083 (sec), leaf count = 301 \[ \left \{\left \{y(x)\to \frac {2 \left (\sqrt {6}-3\right ) \sqrt {-\left (\sqrt {6}-3\right ) a} c_3 \exp \left (-\frac {a x^2}{2}-\sqrt {-\left (\sqrt {6}-3\right ) a} x-\frac {\left (-3+\sqrt {3}+\sqrt {6}\right ) a x}{\sqrt {-\left (\sqrt {6}-3\right ) a}}\right )}{\left (-3-\sqrt {3}+\sqrt {6}\right ) \left (-3+\sqrt {3}+\sqrt {6}\right ) a}+\frac {2 \left (\sqrt {6}-3\right ) \sqrt {-\left (\sqrt {6}-3\right ) a} c_4 \exp \left (-\frac {a x^2}{2}-\sqrt {-\left (\sqrt {6}-3\right ) a} x-\frac {\left (-3-\sqrt {3}+\sqrt {6}\right ) a x}{\sqrt {-\left (\sqrt {6}-3\right ) a}}\right )}{\left (-3-\sqrt {3}+\sqrt {6}\right ) \left (-3+\sqrt {3}+\sqrt {6}\right ) a}+c_1 e^{-\frac {a x^2}{2}-\sqrt {-\left (\sqrt {6}-3\right ) a} x}+c_2 e^{\sqrt {-\left (\sqrt {6}-3\right ) a} x-\frac {a x^2}{2}}\right \}\right \} \]
Maple: cpu = 0.047 (sec), leaf count = 81 \[ \left \{ y \left ( x \right ) ={{\rm e}^{-{\frac {a{x}^{2}}{2}}}} \left ( {\it \_C1}\,{{\rm e}^{-\sqrt {-a\sqrt {6}+3\,a}x}}+{\it \_C2} \,{{\rm e}^{\sqrt {-a\sqrt {6}+3\,a}x}}+{\it \_C3}\,{{\rm e}^{-\sqrt { a\sqrt {6}+3\,a}x}}+{\it \_C4}\,{{\rm e}^{\sqrt {a\sqrt {6}+3\,a}x}} \right ) \right \} \]