\[ y(x) \left (-a^2+x^2 (2 a+2 n+1)+a (-1)^n-x^4\right )+x^2 y''(x)=0 \] ✓ Mathematica : cpu = 0.200588 (sec), leaf count = 260
DSolve[((-1)^n*a - a^2 + (1 + 2*a + 2*n)*x^2 - x^4)*y[x] + x^2*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {c_1 e^{-\frac {x^2}{2}} 2^{\frac {1}{4} \left (\sqrt {4 a^2-4 a (-1)^n+1}+2\right )} \left (x^2\right )^{\frac {1}{4} \left (\sqrt {4 a^2-4 a (-1)^n+1}+2\right )} U\left (\frac {1}{4} \left (-2 a-2 n+\sqrt {4 a^2-4 (-1)^n a+1}+1\right ),\frac {1}{2} \left (\sqrt {4 a^2-4 (-1)^n a+1}+2\right ),x^2\right )}{\sqrt {x}}+\frac {c_2 e^{-\frac {x^2}{2}} 2^{\frac {1}{4} \left (\sqrt {4 a^2-4 a (-1)^n+1}+2\right )} \left (x^2\right )^{\frac {1}{4} \left (\sqrt {4 a^2-4 a (-1)^n+1}+2\right )} L_{\frac {1}{4} \left (2 a+2 n-\sqrt {4 a^2-4 (-1)^n a+1}-1\right )}^{\frac {1}{2} \left (\sqrt {4 a^2-4 (-1)^n a+1}+2\right )-1}\left (x^2\right )}{\sqrt {x}}\right \}\right \}\] ✓ Maple : cpu = 0.156 (sec), leaf count = 71
dsolve(x^2*diff(diff(y(x),x),x)+(-x^4+(2*n+2*a+1)*x^2+(-1)^n*a-a^2)*y(x)=0,y(x))
\[y \left (x \right ) = \frac {\WhittakerW \left (\frac {n}{2}+\frac {a}{2}+\frac {1}{4}, \frac {\sqrt {1-4 \left (-1\right )^{n} a +4 a^{2}}}{4}, x^{2}\right ) c_{2}+\WhittakerM \left (\frac {n}{2}+\frac {a}{2}+\frac {1}{4}, \frac {\sqrt {1-4 \left (-1\right )^{n} a +4 a^{2}}}{4}, x^{2}\right ) c_{1}}{\sqrt {x}}\]