\[ -\left (4 n^2-1\right ) x^2 y''(x)+\left (4 n^2-4 x^4-1\right ) y(x)-\left (4 n^2-1\right ) x y'(x)+x^4 y^{(4)}(x)+4 x^3 y^{(3)}(x)=0 \] ✓ Mathematica : cpu = 1.48031 (sec), leaf count = 232
\[\left \{\left \{y(x)\to c_3 (-1)^{\frac {1}{4} (1-2 n)} 2^{2 n+\frac {1}{2} (2 n-1)-1} x^{1-2 n} \, _0F_3\left (;1-n,1-\frac {n}{2},\frac {3}{2}-\frac {n}{2};\frac {x^4}{64}\right )+c_4 (-1)^{\frac {1}{4} (2 n+1)} 2^{\frac {1}{2} (-2 n-1)-2 n-1} x^{2 n+1} \, _0F_3\left (;\frac {n}{2}+1,\frac {n}{2}+\frac {3}{2},n+1;\frac {x^4}{64}\right )+\frac {\sqrt [4]{-1} c_2 x \, _0F_3\left (;\frac {3}{2},1-\frac {n}{2},\frac {n}{2}+1;\frac {x^4}{64}\right )}{2 \sqrt {2}}-\frac {2 (-1)^{3/4} \sqrt {2} c_1 \, _0F_3\left (;\frac {1}{2},\frac {1}{2}-\frac {n}{2},\frac {n}{2}+\frac {1}{2};\frac {x^4}{64}\right )}{x}\right \}\right \}\] ✓ Maple : cpu = 0.251 (sec), leaf count = 87
\[ \left \{ y \left ( x \right ) ={\frac {1}{x} \left ( {\it \_C4}\,{\mbox {$_0$F$_3$}(\ ;\,{\frac {1}{2}},{\frac {n}{2}}+{\frac {1}{2}},-{\frac {n}{2}}+{\frac {1}{2}};\,{\frac {{x}^{4}}{64}})}+ \left ( {\it \_C3}\,{\mbox {$_0$F$_3$}(\ ;\,{\frac {3}{2}},{\frac {n}{2}}+1,-{\frac {n}{2}}+1;\,{\frac {{x}^{4}}{64}})}+{\it \_C2}\, \left ( {{\rm bei}_{-n}\left (x\right )} \right ) ^{2}+ \left ( {{\rm ber}_{-n}\left (x\right )} \right ) ^{2}{\it \_C2}+{\it \_C1}\, \left ( \left ( {{\rm ber}_{n}\left (x\right )} \right ) ^{2}+ \left ( {{\rm bei}_{n}\left (x\right )} \right ) ^{2} \right ) \right ) {x}^{2} \right ) } \right \} \]