\[ -\left (12 n^2+4 x^4-3\right ) y(x)-\left (4 n^2+3\right ) x^2 y''(x)+\left (12 n^2-3\right ) x y'(x)+x^4 y^{(4)}(x)+4 x^3 y^{(3)}(x)=0 \] ✓ Mathematica : cpu = 1.47867 (sec), leaf count = 196
\[\left \{\left \{y(x)\to \frac {\left (\frac {1}{32}+\frac {i}{32}\right ) \left (8 c_1 x^2 \, _0F_3\left (;\frac {1}{2},\frac {3}{2}-\frac {n}{2},\frac {n}{2}+\frac {3}{2};\frac {x^4}{64}\right )+i \left (c_2 x^4 \, _0F_3\left (;\frac {3}{2},2-\frac {n}{2},\frac {n}{2}+2;\frac {x^4}{64}\right )-8^{2-n} e^{-\frac {1}{2} i \pi n} x^{-2 n} \left (c_3 64^n \, _0F_3\left (;1-n,\frac {1}{2}-\frac {n}{2},-\frac {n}{2};\frac {x^4}{64}\right )+c_4 e^{i \pi n} x^{4 n} \, _0F_3\left (;\frac {n}{2}+\frac {1}{2},\frac {n}{2},n+1;\frac {x^4}{64}\right )\right )\right )\right )}{x}\right \}\right \}\]
✓ Maple : cpu = 0.273 (sec), leaf count = 88
\[ \left \{ y \left ( x \right ) ={\frac {1}{x} \left ( {\it \_C4}\,{x}^{2}{\mbox {$_0$F$_3$}(\ ;\,{\frac {1}{2}},{\frac {n}{2}}+{\frac {3}{2}},{\frac {3}{2}}-{\frac {n}{2}};\,{\frac {{x}^{4}}{64}})}+{\it \_C3}\,{x}^{4}{\mbox {$_0$F$_3$}(\ ;\,{\frac {3}{2}},{\frac {n}{2}}+2,-{\frac {n}{2}}+2;\,{\frac {{x}^{4}}{64}})}+{\it \_C2}\, \left ( {{\rm bei}_{-n}\left (x\right )} \right ) ^{2}+ \left ( {{\rm ber}_{-n}\left (x\right )} \right ) ^{2}{\it \_C2}+{\it \_C1}\, \left ( \left ( {{\rm ber}_{n}\left (x\right )} \right ) ^{2}+ \left ( {{\rm bei}_{n}\left (x\right )} \right ) ^{2} \right ) \right ) } \right \} \]