\[ -\frac {1}{16} b^4 y(x)+x (2 n-2 \nu +4) y^{(3)}(x)+(n-\nu +1) (n-\nu +2) y''(x)+x^2 y^{(4)}(x)=0 \] ✓ Mathematica : cpu = 0.114711 (sec), leaf count = 319
DSolve[-1/16*(b^4*y[x]) + (1 + n - nu)*(2 + n - nu)*Derivative[2][y][x] + (4 + 2*n - 2*nu)*x*Derivative[3][y][x] + x^2*Derivative[4][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_4 i^{-n+\nu +1} 2^{3 n-3 \nu -3} b^{2 (-n+\nu +1)+n-\nu -2} x^{\frac {1}{2} (n-\nu -2)-n+\nu +1} \Gamma (-n+\nu +2) \left (I_{\nu -n}\left (b \sqrt {x}\right )-J_{\nu -n}\left (b \sqrt {x}\right )\right )+c_3 i^{\nu -n} 2^{3 n-3 \nu -1} b^{2 (\nu -n)+n-\nu } x^{\frac {n-\nu }{2}-n+\nu } \Gamma (-n+\nu +1) \left (J_{\nu -n}\left (b \sqrt {x}\right )+I_{\nu -n}\left (b \sqrt {x}\right )\right )+i c_2 2^{n-\nu -3} b^{\nu -n} x^{\frac {1}{2} (-n+\nu -2)+1} \Gamma (n-\nu +2) \left (I_{n-\nu }\left (b \sqrt {x}\right )-J_{n-\nu }\left (b \sqrt {x}\right )\right )+c_1 2^{n-\nu -1} b^{\nu -n} x^{\frac {\nu -n}{2}} \Gamma (n-\nu +1) \left (J_{n-\nu }\left (b \sqrt {x}\right )+I_{n-\nu }\left (b \sqrt {x}\right )\right )\right \}\right \}\] ✓ Maple : cpu = 0.123 (sec), leaf count = 67
dsolve(x^2*diff(diff(diff(diff(y(x),x),x),x),x)+(2*n-2*nu+4)*x*diff(diff(diff(y(x),x),x),x)+(n-nu+1)*(n-nu+2)*diff(diff(y(x),x),x)-1/16*b^4*y(x)=0,y(x))
\[y \left (x \right ) = x^{-\frac {n}{2}+\frac {\nu }{2}} \left (\BesselK \left (n -\nu , b \sqrt {x}\right ) c_{3}+\BesselI \left (n -\nu , b \sqrt {x}\right ) c_{1}+\BesselJ \left (n -\nu , b \sqrt {x}\right ) c_{2}+\BesselY \left (n -\nu , b \sqrt {x}\right ) c_{4}\right )\]