\[ \boxed { {x}^{4}{\it d4y} \left ( x \right ) -2\,n \left ( n+1 \right ) {x}^{2}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) +4\,n \left ( n+1 \right ) x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( a{x}^{4}+n \left ( n+1 \right ) \left ( n+3 \right ) \left ( n-2 \right ) \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 3.736975 (sec), leaf count = 400 \[ \left \{\left \{y(x)\to c_1 \left (-2^{n-\frac {5}{2}}\right ) \sqrt {x} a^{\frac {2-n}{4}+\frac {1}{4} \left (n-\frac {3}{2}\right )} \Gamma \left (\frac {3}{2}-n\right ) \left (\cos \left (\frac {3}{4} \pi \left (\frac {3}{2}-n\right )\right ) \text {ber}_{-n-\frac {1}{2}}\left (\sqrt [4]{a} x\right )+\sin \left (\frac {3}{4} \pi \left (\frac {3}{2}-n\right )\right ) \text {bei}_{-n-\frac {1}{2}}\left (\sqrt [4]{a} x\right )\right )+c_2 2^{n-\frac {1}{2}} \sqrt {x} a^{\frac {1}{4} \left (n+\frac {1}{2}\right )-\frac {n}{4}} \Gamma \left (\frac {1}{2}-n\right ) \left (\cos \left (\frac {3}{4} \pi \left (-n-\frac {1}{2}\right )\right ) \text {ber}_{-n-\frac {1}{2}}\left (\sqrt [4]{a} x\right )+\sin \left (\frac {3}{4} \pi \left (-n-\frac {1}{2}\right )\right ) \text {bei}_{-n-\frac {1}{2}}\left (\sqrt [4]{a} x\right )\right )+c_3 2^{-n-\frac {3}{2}} \sqrt {x} a^{\frac {1}{4} \left (-n-\frac {1}{2}\right )+\frac {n+1}{4}} \Gamma \left (n+\frac {3}{2}\right ) \left (\cos \left (\frac {3}{4} \pi \left (n+\frac {1}{2}\right )\right ) \text {ber}_{n+\frac {1}{2}}\left (\sqrt [4]{a} x\right )+\sin \left (\frac {3}{4} \pi \left (n+\frac {1}{2}\right )\right ) \text {bei}_{n+\frac {1}{2}}\left (\sqrt [4]{a} x\right )\right )-c_4 2^{-n-\frac {7}{2}} \sqrt {x} a^{\frac {1}{4} \left (-n-\frac {5}{2}\right )+\frac {n+3}{4}} \Gamma \left (n+\frac {5}{2}\right ) \left (\cos \left (\frac {3}{4} \pi \left (n+\frac {5}{2}\right )\right ) \text {ber}_{n+\frac {1}{2}}\left (\sqrt [4]{a} x\right )+\sin \left (\frac {3}{4} \pi \left (n+\frac {5}{2}\right )\right ) \text {bei}_{n+\frac {1}{2}}\left (\sqrt [4]{a} x\right )\right )\right \}\right \} \]
Maple: cpu = 0.140 (sec), leaf count = 77 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,\sqrt {x}{{\sl J}_{n+{\frac { 1}{2}}}\left (\sqrt [4]{-a}x\right )}+{\it \_C2}\,\sqrt {x}{{\sl Y}_{n+{ \frac {1}{2}}}\left (\sqrt [4]{-a}x\right )}+{\it \_C3}\,\sqrt {x}{ {\sl J}_{n+{\frac {1}{2}}}\left (\sqrt {-\sqrt {-a}}x\right )}+{\it \_C4 }\,\sqrt {x}{{\sl Y}_{n+{\frac {1}{2}}}\left (\sqrt {-\sqrt {-a}}x \right )} \right \} \]