\[ \left (a x^2+6 n\right ) y'(x)-2 a x y(x)-2 (n+1) x y''(x)+x^2 y^{(3)}(x)=0 \] ✓ Mathematica : cpu = 12.9682 (sec), leaf count = 701
\[\left \{\left \{y(x)\to \frac {\pi c_3 2^{-n-\frac {3}{2}} x \left (\sqrt {a} x\right )^{-n-\frac {1}{2}} \left (a^{3/2} 2^{2 n} x^3 \csc \left (\frac {1}{2} \pi (2 n+1)\right ) \Gamma \left (\frac {3}{2}-n\right ) \Gamma \left (n+\frac {3}{2}\right ) \Gamma \left (\frac {1}{2} (2 n+3)\right ) J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right ) \, _1\tilde {F}_2\left (\frac {3}{2}-n;\frac {1}{2} (-2 n-1)+1,\frac {5}{2}-n;-\frac {a x^2}{4}\right )+8 n^2 \Gamma \left (\frac {1}{2} (2 n+3)\right ) \left (\sqrt {a} x\right )^{2 n} Y_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )-8 n^2 \cot \left (\frac {1}{2} \pi (2 n+1)\right ) \Gamma \left (n+\frac {3}{2}\right ) \left (\sqrt {a} x\right )^{2 n} J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+a 2^{n+\frac {1}{2}} x^2 \cot \left (\frac {1}{2} \pi (2 n+1)\right ) \Gamma \left (n+\frac {3}{2}\right )^2 \left (\sqrt {a} x\right )^{n+\frac {1}{2}} J_{n-\frac {1}{2}}\left (\sqrt {a} x\right ) J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )-a 2^{n+\frac {1}{2}} x^2 \Gamma \left (n+\frac {3}{2}\right ) \Gamma \left (\frac {1}{2} (2 n+3)\right ) \left (\sqrt {a} x\right )^{n+\frac {1}{2}} J_{n-\frac {1}{2}}\left (\sqrt {a} x\right ) Y_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )-2 \Gamma \left (\frac {1}{2} (2 n+3)\right ) \left (\sqrt {a} x\right )^{2 n} Y_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+2 \cot \left (\frac {1}{2} \pi (2 n+1)\right ) \Gamma \left (n+\frac {3}{2}\right ) \left (\sqrt {a} x\right )^{2 n} J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+\sqrt {a} 2^{n+\frac {3}{2}} x \cot \left (\frac {1}{2} \pi (2 n+1)\right ) \Gamma \left (n+\frac {3}{2}\right )^2 \left (\sqrt {a} x\right )^{n+\frac {1}{2}} J_{n-\frac {3}{2}}\left (\sqrt {a} x\right ) J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )-\sqrt {a} 2^{n+\frac {3}{2}} x \Gamma \left (n+\frac {3}{2}\right ) \Gamma \left (\frac {1}{2} (2 n+3)\right ) \left (\sqrt {a} x\right )^{n+\frac {1}{2}} J_{n-\frac {3}{2}}\left (\sqrt {a} x\right ) Y_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )\right )}{a^{3/2} \Gamma \left (n+\frac {3}{2}\right ) \Gamma \left (\frac {1}{2} (2 n+3)\right )}+c_1 x^{\frac {1}{2} (2 n+1)} J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+c_2 x^{\frac {1}{2} (2 n+1)} Y_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )\right \}\right \}\] ✓ Maple : cpu = 0.262 (sec), leaf count = 53
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{x}^{n+{\frac {1}{2}}}{{\sl J}_{-n-{\frac {1}{2}}}\left (\sqrt {a}x\right )}+{\it \_C2}\,{x}^{n+{\frac {1}{2}}}{{\sl Y}_{-n-{\frac {1}{2}}}\left (\sqrt {a}x\right )}+{\it \_C3}\, \left ( a{x}^{2}+4\,n-2 \right ) \right \} \]