\[ \boxed { {x}^{2}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( -{x}^{4}+ \left ( 2\,n+2\,a+1 \right ) {x}^{2}+ \left ( -1 \right ) ^{n}a-{a}^{2} \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.289037 (sec), leaf count = 260 \[ \left \{\left \{y(x)\to \frac {c_1 e^{-\frac {x^2}{2}} 2^{\frac {1}{4} \left (\sqrt {4 a^2-4 a (-1)^n+1}+2\right )} \left (x^2\right )^{\frac {1}{4} \left (\sqrt {4 a^2-4 a (-1)^n+1}+2\right )} U\left (\frac {1}{4} \left (-2 a-2 n+\sqrt {4 a^2-4 (-1)^n a+1}+1\right ),\frac {1}{2} \left (\sqrt {4 a^2-4 (-1)^n a+1}+2\right ),x^2\right )}{\sqrt {x}}+\frac {c_2 e^{-\frac {x^2}{2}} 2^{\frac {1}{4} \left (\sqrt {4 a^2-4 a (-1)^n+1}+2\right )} \left (x^2\right )^{\frac {1}{4} \left (\sqrt {4 a^2-4 a (-1)^n+1}+2\right )} L_{\frac {1}{4} \left (-\sqrt {4 a^2-4 a (-1)^n+1}+2 a+2 n-1\right )}^{\frac {1}{2} \left (\sqrt {4 a^2-4 a (-1)^n+1}+2\right )-1}\left (x^2\right )}{\sqrt {x}}\right \}\right \} \]
Maple: cpu = 0.390 (sec), leaf count = 73 \[ \left \{ y \left ( x \right ) ={{\it \_C1}{{\sl M}_{{\frac {n}{2}}+{ \frac {a}{2}}+{\frac {1}{4}},\,{\frac {1}{4}\sqrt {1-4\, \left ( -1 \right ) ^{n}a+4\,{a}^{2}}}}\left ({x}^{2}\right )}{\frac {1}{\sqrt {x}} }}+{{\it \_C2}{{\sl W}_{{\frac {n}{2}}+{\frac {a}{2}}+{\frac {1}{4}} ,\,{\frac {1}{4}\sqrt {1-4\, \left ( -1 \right ) ^{n}a+4\,{a}^{2}}} }\left ({x}^{2}\right )}{\frac {1}{\sqrt {x}}}} \right \} \]