\[ y(x) \left (a x^4+(n-2) n (n+1) (n+3)\right )-2 n (n+1) x^2 y''(x)+4 n (n+1) x y'(x)+x^4 y^{(4)}(x)=0 \] ✓ Mathematica : cpu = 2.04031 (sec), leaf count = 400
DSolve[((-2 + n)*n*(1 + n)*(3 + n) + a*x^4)*y[x] + 4*n*(1 + n)*x*Derivative[1][y][x] - 2*n*(1 + n)*x^2*Derivative[2][y][x] + x^4*Derivative[4][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 \left (-2^{n-\frac {5}{2}}\right ) \sqrt {x} a^{\frac {2-n}{4}+\frac {1}{4} \left (n-\frac {3}{2}\right )} \operatorname {Gamma}\left (\frac {3}{2}-n\right ) \left (\cos \left (\frac {3}{4} \pi \left (\frac {3}{2}-n\right )\right ) \text {ber}_{-n-\frac {1}{2}}\left (\sqrt [4]{a} x\right )+\sin \left (\frac {3}{4} \pi \left (\frac {3}{2}-n\right )\right ) \text {bei}_{-n-\frac {1}{2}}\left (\sqrt [4]{a} x\right )\right )+c_2 2^{n-\frac {1}{2}} \sqrt {x} a^{\frac {1}{4} \left (n+\frac {1}{2}\right )-\frac {n}{4}} \operatorname {Gamma}\left (\frac {1}{2}-n\right ) \left (\cos \left (\frac {3}{4} \pi \left (-n-\frac {1}{2}\right )\right ) \text {ber}_{-n-\frac {1}{2}}\left (\sqrt [4]{a} x\right )+\sin \left (\frac {3}{4} \pi \left (-n-\frac {1}{2}\right )\right ) \text {bei}_{-n-\frac {1}{2}}\left (\sqrt [4]{a} x\right )\right )+c_3 2^{-n-\frac {3}{2}} \sqrt {x} a^{\frac {1}{4} \left (-n-\frac {1}{2}\right )+\frac {n+1}{4}} \operatorname {Gamma}\left (n+\frac {3}{2}\right ) \left (\cos \left (\frac {3}{4} \pi \left (n+\frac {1}{2}\right )\right ) \text {ber}_{n+\frac {1}{2}}\left (\sqrt [4]{a} x\right )+\sin \left (\frac {3}{4} \pi \left (n+\frac {1}{2}\right )\right ) \text {bei}_{n+\frac {1}{2}}\left (\sqrt [4]{a} x\right )\right )-c_4 2^{-n-\frac {7}{2}} \sqrt {x} a^{\frac {1}{4} \left (-n-\frac {5}{2}\right )+\frac {n+3}{4}} \operatorname {Gamma}\left (n+\frac {5}{2}\right ) \left (\cos \left (\frac {3}{4} \pi \left (n+\frac {5}{2}\right )\right ) \text {ber}_{n+\frac {1}{2}}\left (\sqrt [4]{a} x\right )+\sin \left (\frac {3}{4} \pi \left (n+\frac {5}{2}\right )\right ) \text {bei}_{n+\frac {1}{2}}\left (\sqrt [4]{a} x\right )\right )\right \}\right \}\] ✓ Maple : cpu = 0.262 (sec), leaf count = 69
dsolve(x^4*diff(diff(diff(diff(y(x),x),x),x),x)-2*n*(n+1)*x^2*diff(diff(y(x),x),x)+4*n*(n+1)*x*diff(y(x),x)+(a*x^4+n*(n+1)*(n+3)*(n-2))*y(x)=0,y(x))
\[y \left (x \right ) = \sqrt {x}\, \left (\operatorname {BesselJ}\left (n +\frac {1}{2}, \sqrt {-\sqrt {-a}}\, x \right ) c_{3}+\operatorname {BesselY}\left (n +\frac {1}{2}, \sqrt {-\sqrt {-a}}\, x \right ) c_{4}+\operatorname {BesselJ}\left (n +\frac {1}{2}, \left (-a \right )^{\frac {1}{4}} x \right ) c_{1}+\operatorname {BesselY}\left (n +\frac {1}{2}, \left (-a \right )^{\frac {1}{4}} x \right ) c_{2}\right )\]