\[ \left (a x^2+6 n\right ) y'(x)-2 a x y(x)-2 (n+1) x y''(x)+x^2 y^{(3)}(x)=0 \] ✓ Mathematica : cpu = 8.40185 (sec), leaf count = 584
DSolve[-2*a*x*y[x] + (6*n + a*x^2)*Derivative[1][y][x] - 2*(1 + n)*x*Derivative[2][y][x] + x^2*Derivative[3][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to -\frac {\pi c_3 2^{-n-\frac {3}{2}} x \left (\sqrt {a} x\right )^{-n-\frac {1}{2}} \left (-a^{3/2} 2^{2 n} x^3 \sec (\pi n) \Gamma \left (\frac {3}{2}-n\right ) \Gamma \left (n+\frac {3}{2}\right ) J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right ) \, _1\tilde {F}_2\left (\frac {3}{2}-n;\frac {1}{2}-n,\frac {5}{2}-n;-\frac {a x^2}{4}\right )-8 n^2 \left (\sqrt {a} x\right )^{2 n} Y_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )-8 n^2 \tan (\pi n) \left (\sqrt {a} x\right )^{2 n} J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+a 2^{n+\frac {1}{2}} x^2 \tan (\pi n) \Gamma \left (n+\frac {3}{2}\right ) \left (\sqrt {a} x\right )^{n+\frac {1}{2}} J_{n-\frac {1}{2}}\left (\sqrt {a} x\right ) J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+a 2^{n+\frac {1}{2}} x^2 \Gamma \left (n+\frac {3}{2}\right ) \left (\sqrt {a} x\right )^{n+\frac {1}{2}} J_{n-\frac {1}{2}}\left (\sqrt {a} x\right ) Y_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+2 \left (\sqrt {a} x\right )^{2 n} Y_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+2 \tan (\pi n) \left (\sqrt {a} x\right )^{2 n} J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+\sqrt {a} 2^{n+\frac {3}{2}} x \tan (\pi n) \Gamma \left (n+\frac {3}{2}\right ) \left (\sqrt {a} x\right )^{n+\frac {1}{2}} J_{n-\frac {3}{2}}\left (\sqrt {a} x\right ) J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+\sqrt {a} 2^{n+\frac {3}{2}} x \Gamma \left (n+\frac {3}{2}\right ) \left (\sqrt {a} x\right )^{n+\frac {1}{2}} J_{n-\frac {3}{2}}\left (\sqrt {a} x\right ) Y_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )\right )}{a^{3/2} \Gamma \left (n+\frac {3}{2}\right )}+c_1 x^{\frac {1}{2} (2 n+1)} J_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )+c_2 x^{\frac {1}{2} (2 n+1)} Y_{\frac {1}{2} (2 n+1)}\left (\sqrt {a} x\right )\right \}\right \}\] ✓ Maple : cpu = 0.197 (sec), leaf count = 53
dsolve(x^2*diff(diff(diff(y(x),x),x),x)-2*(n+1)*x*diff(diff(y(x),x),x)+(a*x^2+6*n)*diff(y(x),x)-2*a*x*y(x)=0,y(x))
\[y \left (x \right ) = c_{1} x^{n +\frac {1}{2}} \BesselJ \left (-n -\frac {1}{2}, \sqrt {a}\, x \right )+c_{2} x^{n +\frac {1}{2}} \BesselY \left (-n -\frac {1}{2}, \sqrt {a}\, x \right )+c_{3} \left (a \,x^{2}+4 n -2\right )\]