\[ \left (-12 n^2-4 x^4+3\right ) y(x)-\left (4 n^2+3\right ) x^2 y''(x)+\left (12 n^2-3\right ) x y'(x)+x^4 y^{(4)}(x)+4 x^3 y^{(3)}(x)=0 \] ✓ Mathematica : cpu = 1.00688 (sec), leaf count = 230
\[\left \{\left \{y(x)\to \frac {\sqrt [4]{-1} c_1 x \, _0F_3\left (;\frac {1}{2},\frac {3}{2}-\frac {n}{2},\frac {n}{2}+\frac {3}{2};\frac {x^4}{64}\right )}{2 \sqrt {2}}+c_3 (-1)^{\frac {1}{4} (-2 n-1)} 2^{2 n+\frac {1}{2} (2 n+1)+1} x^{-2 n-1} \, _0F_3\left (;1-n,\frac {1}{2}-\frac {n}{2},-\frac {n}{2};\frac {x^4}{64}\right )+c_4 (-1)^{\frac {1}{4} (2 n-1)} 2^{\frac {1}{2} (1-2 n)-2 n+1} x^{2 n-1} \, _0F_3\left (;\frac {n}{2}+\frac {1}{2},\frac {n}{2},n+1;\frac {x^4}{64}\right )+\frac {(-1)^{3/4} c_2 x^3 \, _0F_3\left (;\frac {3}{2},2-\frac {n}{2},\frac {n}{2}+2;\frac {x^4}{64}\right )}{16 \sqrt {2}}\right \}\right \}\] ✓ Maple : cpu = 0.203 (sec), leaf count = 88
\[y \relax (x ) = \frac {c_{4} x^{2} \hypergeom \left (\left [\right ], \left [\frac {1}{2}, \frac {3}{2}-\frac {n}{2}, \frac {n}{2}+\frac {3}{2}\right ], \frac {x^{4}}{64}\right )+c_{3} x^{4} \hypergeom \left (\left [\right ], \left [\frac {3}{2}, -\frac {n}{2}+2, \frac {n}{2}+2\right ], \frac {x^{4}}{64}\right )+c_{2} \mathit {bei}_{-n}\relax (x )^{2}+\mathit {ber}_{-n}\relax (x )^{2} c_{2}+c_{1} \left (\mathit {ber}_{n}\relax (x )^{2}+\mathit {bei}_{n}\relax (x )^{2}\right )}{x}\]