ODE
\[ -y(x) \left (4 a x^2+n^2-x^4\right )+x^2 y''(x)+x y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.131236 (sec), leaf count = 93
\[\left \{\left \{y(x)\to \frac {2^{\frac {n+1}{2}} e^{-\frac {i x^2}{2}} \left (x^2\right )^{\frac {n+1}{2}} \left (c_1 U\left (\frac {1}{2} (-2 i a+n+1),n+1,i x^2\right )+c_2 L_{i a-\frac {n}{2}-\frac {1}{2}}^n\left (i x^2\right )\right )}{x}\right \}\right \}\]
Maple ✓
cpu = 0.092 (sec), leaf count = 41
\[ \left \{ y \left ( x \right ) ={\frac {1}{x} \left ( {\it \_C2}\,{{\sl W}_{ia,\,{\frac {n}{2}}}\left (i{x}^{2}\right )}+{\it \_C1}\,{{\sl M}_{ia,\,{\frac {n}{2}}}\left (i{x}^{2}\right )} \right ) } \right \} \] Mathematica raw input
DSolve[-((n^2 + 4*a*x^2 - x^4)*y[x]) + x*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (2^((1 + n)/2)*(x^2)^((1 + n)/2)*(C[1]*HypergeometricU[(1 - (2*I)*a + n)/2, 1 + n, I*x^2] + C[2]*LaguerreL[-1/2 + I*a - n/2, n, I*x^2]))/(E^((I/2)*x^2)*x)}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)-(-x^4+4*a*x^2+n^2)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (_C2*WhittakerW(I*a,1/2*n,I*x^2)+_C1*WhittakerM(I*a,1/2*n,I*x^2))/x