\[ \boxed { {x}^{2}{\it d4y} \left ( x \right ) + \left ( 2\,n-2\,\nu +4 \right ) x{\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}y \left ( x \right ) + \left ( n-\nu +1 \right ) \left ( n-\nu +2 \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) -1/16\,{b}^{4}y \left ( x \right ) =0} \]
Mathematica: cpu = 0.160520 (sec), leaf count = 319 \[ \left \{\left \{y(x)\to c_4 i^{-n+\nu +1} 2^{3 n-3 \nu -3} b^{2 (-n+\nu +1)+n-\nu -2} x^{\frac {1}{2} (n-\nu -2)-n+\nu +1} \Gamma (-n+\nu +2) \left (I_{\nu -n}\left (b \sqrt {x}\right )-J_{\nu -n}\left (b \sqrt {x}\right )\right )+c_3 i^{\nu -n} 2^{3 n-3 \nu -1} b^{2 (\nu -n)+n-\nu } x^{\frac {n-\nu }{2}-n+\nu } \Gamma (-n+\nu +1) \left (J_{\nu -n}\left (b \sqrt {x}\right )+I_{\nu -n}\left (b \sqrt {x}\right )\right )+i c_2 2^{n-\nu -3} b^{\nu -n} x^{\frac {1}{2} (-n+\nu -2)+1} \Gamma (n-\nu +2) \left (I_{n-\nu }\left (b \sqrt {x}\right )-J_{n-\nu }\left (b \sqrt {x}\right )\right )+c_1 2^{n-\nu -1} b^{\nu -n} x^{\frac {\nu -n}{2}} \Gamma (n-\nu +1) \left (J_{n-\nu }\left (b \sqrt {x}\right )+I_{n-\nu }\left (b \sqrt {x}\right )\right )\right \}\right \} \]
Maple: cpu = 0.062 (sec), leaf count = 93 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{x}^{-{\frac {n}{2}}+{\frac { \nu }{2}}}{{\sl I}_{n-\nu }\left (b\sqrt {x}\right )}+{\it \_C2}\,{x}^{-{ \frac {n}{2}}+{\frac {\nu }{2}}}{{\sl J}_{n-\nu }\left (b\sqrt {x} \right )}+{\it \_C3}\,{x}^{-{\frac {n}{2}}+{\frac {\nu }{2}}}{{\sl K}_{n -\nu }\left (b\sqrt {x}\right )}+{\it \_C4}\,{x}^{-{\frac {n}{2}}+{\frac {\nu }{2}}}{{\sl Y}_{n-\nu }\left (b\sqrt {x}\right )} \right \} \]