ODE
\[ x^2 y''(x)-y(x) \left ((n-1) n-a^2 x^2\right )=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.176109 (sec), leaf count = 36
\[\left \{\left \{y(x)\to \sqrt {x} \left (c_1 J_{n-\frac {1}{2}}(a x)+c_2 Y_{n-\frac {1}{2}}(a x)\right )\right \}\right \}\]
Maple ✓
cpu = 0.081 (sec), leaf count = 29
\[\left [y \left (x \right ) = \textit {\_C1} \sqrt {x}\, \BesselJ \left (n -\frac {1}{2}, a x \right )+\textit {\_C2} \sqrt {x}\, \BesselY \left (n -\frac {1}{2}, a x \right )\right ]\] Mathematica raw input
DSolve[-(((-1 + n)*n - a^2*x^2)*y[x]) + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> Sqrt[x]*(BesselJ[-1/2 + n, a*x]*C[1] + BesselY[-1/2 + n, a*x]*C[2])}}
Maple raw input
dsolve(x^2*diff(diff(y(x),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)]