\[ \boxed { {x}^{2}{\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}y \left ( x \right ) -2\, \left ( n+1 \right ) x{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( a{x}^{2}+6\,n \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -2\,axy \left ( x \right ) =0} \]
Mathematica: cpu = 8.482077 (sec), leaf count = 584 \[ \left \{\left \{y(x)\to -\frac {\pi c_3 2^{-n-\frac {3}{2}} x \left (\sqrt {a} x\right )^{-n-\frac {1}{2}} \left (-a^{3/2} 2^{2 n} x^3 \sec (\pi n) \Gamma \left (\frac {3}{2}-n\right ) \Gamma \left (n+\frac {3}{2}\right ) J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right ) \, _1\tilde {F}_2\left (\frac {3}{2}-n;\frac {1}{2}-n,\frac {5}{2}-n;-\frac {a x^2}{4}\right )-8 n^2 \left (\sqrt {a} x\right )^{2 n} Y_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )-8 n^2 \tan (\pi n) \left (\sqrt {a} x\right )^{2 n} J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+a 2^{n+\frac {1}{2}} x^2 \tan (\pi n) \Gamma \left (n+\frac {3}{2}\right ) \left (\sqrt {a} x\right )^{n+\frac {1}{2}} J_{n-\frac {1}{2}}\left (\sqrt {a} x\right ) J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+a 2^{n+\frac {1}{2}} x^2 \Gamma \left (n+\frac {3}{2}\right ) \left (\sqrt {a} x\right )^{n+\frac {1}{2}} J_{n-\frac {1}{2}}\left (\sqrt {a} x\right ) Y_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+2 \left (\sqrt {a} x\right )^{2 n} Y_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+2 \tan (\pi n) \left (\sqrt {a} x\right )^{2 n} J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+\sqrt {a} 2^{n+\frac {3}{2}} x \tan (\pi n) \Gamma \left (n+\frac {3}{2}\right ) \left (\sqrt {a} x\right )^{n+\frac {1}{2}} J_{n-\frac {3}{2}}\left (\sqrt {a} x\right ) J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+\sqrt {a} 2^{n+\frac {3}{2}} x \Gamma \left (n+\frac {3}{2}\right ) \left (\sqrt {a} x\right )^{n+\frac {1}{2}} J_{n-\frac {3}{2}}\left (\sqrt {a} x\right ) Y_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )\right )}{a^{3/2} \Gamma \left (n+\frac {3}{2}\right )}+c_1 x^{\frac {1}{2} (2 n+1)} J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+c_2 x^{\frac {1}{2} (2 n+1)} Y_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )\right \}\right \} \]
Maple: cpu = 0.234 (sec), leaf count = 53 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{x}^{n+{\frac {1}{2}}}{ {\sl J}_{-n-{\frac {1}{2}}}\left (\sqrt {a}x\right )}+{\it \_C2}\,{x}^{n +{\frac {1}{2}}}{{\sl Y}_{-n-{\frac {1}{2}}}\left (\sqrt {a}x\right )}+{ \it \_C3}\, \left ( a{x}^{2}+4\,n-2 \right ) \right \} \]