\[ \boxed { {x}^{4}{\it d4y} \left ( x \right ) +4\,{x}^{3}{\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}y \left ( x \right ) - \left ( 4\,{n}^{2}-1 \right ) {x}^{2}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) - \left ( 4\,{n}^{2}-1 \right ) x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( -4\,{x}^{4}+4\,{n}^{2}-1 \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 1.852235 (sec), leaf count = 232 \[ \left \{\left \{y(x)\to \frac {\sqrt [4]{-1} c_2 x \, _0F_3\left (;\frac {3}{2},1-\frac {n}{2},\frac {n}{2}+1;\frac {x^4}{64}\right )}{2 \sqrt {2}}-\frac {2 (-1)^{3/4} \sqrt {2} c_1 \, _0F_3\left (;\frac {1}{2},\frac {1}{2}-\frac {n}{2},\frac {n}{2}+\frac {1}{2};\frac {x^4}{64}\right )}{x}+c_3 (-1)^{\frac {1}{4} (1-2 n)} 2^{2 n+\frac {1}{2} (2 n-1)-1} x^{1-2 n} \, _0F_3\left (;1-n,1-\frac {n}{2},\frac {3}{2}-\frac {n}{2};\frac {x^4}{64}\right )+c_4 (-1)^{\frac {1}{4} (2 n+1)} 2^{\frac {1}{2} (-2 n-1)-2 n-1} x^{2 n+1} \, _0F_3\left (;\frac {n}{2}+1,\frac {n}{2}+\frac {3}{2},n+1;\frac {x^4}{64}\right )\right \}\right \} \]
Maple: cpu = 0.110 (sec), leaf count = 83 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,x \left ( \left ( {{\rm ber}_{ n}\left (x\right )} \right ) ^{2}+ \left ( {{\rm bei}_{n}\left (x\right )} \right ) ^{2} \right ) +{\it \_C2}\,x \left ( \left ( {{\rm ber}_{-n }\left (x\right )} \right ) ^{2}+ \left ( {{\rm bei}_{-n}\left (x\right )} \right ) ^{2} \right ) +{\it \_C3}\,x {\mbox {$_0$F$_3$}(\ ;\,{\frac {3}{2}},1-{\frac {n}{2}},1+{\frac {n}{2}};\,{\frac {{x}^{4}}{64}})} +{\frac {{\it \_C4}}{x} {\mbox {$_0$F$_3$}(\ ;\,{\frac {1}{2}},{\frac {1}{2}}+{\frac {n}{2}},{\frac {1}{2}}-{\frac {n}{2}};\,{\frac {{x}^{4}}{64}})} } \right \} \]