ODE
\[ -y(x) \left (n (n+1)-a^2 x^2\right )+x^2 y''(x)+2 x y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.170451 (sec), leaf count = 22
\[\{\{y(x)\to c_1 j_n(a x)+c_2 y_n(a x)\}\}\]
Maple ✓
cpu = 0.257 (sec), leaf count = 29
\[\left [y \left (x \right ) = \frac {\textit {\_C1} \BesselJ \left (n +\frac {1}{2}, a x \right )}{\sqrt {x}}+\frac {\textit {\_C2} \BesselY \left (n +\frac {1}{2}, a x \right )}{\sqrt {x}}\right ]\] Mathematica raw input
DSolve[-((n*(1 + n) - a^2*x^2)*y[x]) + 2*x*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*SphericalBesselJ[n, a*x] + C[2]*SphericalBesselY[n, a*x]}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+2*x*diff(y(x),x)-(n*(n+1)-a^2*x^2)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1/x^(1/2)*BesselJ(n+1/2,a*x)+_C2/x^(1/2)*BesselY(n+1/2,a*x)]