\[ -\left (4 n^2+3\right ) x^2 y''(x)-\left (12 n^2+4 x^4-3\right ) y(x)+\left (12 n^2-3\right ) x y'(x)+x^4 y^{(4)}(x)+4 x^3 y^{(3)}(x)=0 \] ✓ Mathematica : cpu = 0.988763 (sec), leaf count = 230
\[\left \{\left \{y(x)\to c_3 (-1)^{\frac {1}{4} (-2 n-1)} 2^{2 n+\frac {1}{2} (2 n+1)+1} x^{-2 n-1} \, _0F_3\left (;1-n,\frac {1}{2}-\frac {n}{2},-\frac {n}{2};\frac {x^4}{64}\right )+c_4 (-1)^{\frac {1}{4} (2 n-1)} 2^{\frac {1}{2} (1-2 n)-2 n+1} x^{2 n-1} \, _0F_3\left (;\frac {n}{2}+\frac {1}{2},\frac {n}{2},n+1;\frac {x^4}{64}\right )+\frac {(-1)^{3/4} c_2 x^3 \, _0F_3\left (;\frac {3}{2},2-\frac {n}{2},\frac {n}{2}+2;\frac {x^4}{64}\right )}{16 \sqrt {2}}+\frac {\sqrt [4]{-1} c_1 x \, _0F_3\left (;\frac {1}{2},\frac {3}{2}-\frac {n}{2},\frac {n}{2}+\frac {3}{2};\frac {x^4}{64}\right )}{2 \sqrt {2}}\right \}\right \}\] ✓ Maple : cpu = 0.227 (sec), leaf count = 88
\[ \left \{ y \left ( x \right ) ={\frac {1}{x} \left ( {\it \_C4}\,{x}^{2}{\mbox {$_0$F$_3$}(\ ;\,{\frac {1}{2}},{\frac {n}{2}}+{\frac {3}{2}},{\frac {3}{2}}-{\frac {n}{2}};\,{\frac {{x}^{4}}{64}})}+{\it \_C3}\,{x}^{4}{\mbox {$_0$F$_3$}(\ ;\,{\frac {3}{2}},{\frac {n}{2}}+2,-{\frac {n}{2}}+2;\,{\frac {{x}^{4}}{64}})}+{\it \_C2}\, \left ( {{\rm bei}_{-n}\left (x\right )} \right ) ^{2}+ \left ( {{\rm ber}_{-n}\left (x\right )} \right ) ^{2}{\it \_C2}+{\it \_C1}\, \left ( \left ( {{\rm ber}_{n}\left (x\right )} \right ) ^{2}+ \left ( {{\rm bei}_{n}\left (x\right )} \right ) ^{2} \right ) \right ) } \right \} \]