\[ y(x) \left (a x^4+(n-2) n (n+1) (n+3)\right )-2 n (n+1) x^2 y''(x)+4 n (n+1) x y'(x)+x^4 y^{(4)}(x)=0 \] ✓ Mathematica : cpu = 2.85719 (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 = 0.26 (sec), leaf count = 69
\[y \relax (x ) = \sqrt {x}\, \left (\BesselY \left (n +\frac {1}{2}, \left (-a \right )^{\frac {1}{4}} x \right ) c_{2}+\BesselJ \left (n +\frac {1}{2}, \left (-a \right )^{\frac {1}{4}} x \right ) c_{1}+\BesselY \left (n +\frac {1}{2}, \sqrt {-\sqrt {-a}}\, x \right ) c_{4}+\BesselJ \left (n +\frac {1}{2}, \sqrt {-\sqrt {-a}}\, x \right ) c_{3}\right )\]