ODE
\[ x^2 y''(x)-y(x) \left (a^2 x^2+n (n+1)\right )=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0290829 (sec), leaf count = 42
\[\left \{\left \{y(x)\to \sqrt {x} \left (c_1 J_{n+\frac {1}{2}}(-i a x)+c_2 Y_{n+\frac {1}{2}}(-i a x)\right )\right \}\right \}\]
Maple ✓
cpu = 0.038 (sec), leaf count = 39
\[ \left \{ y \left ( x \right ) =\sqrt {x} \left ( {{\sl Y}_{n+{\frac {1}{2}}}\left (\sqrt {-{a}^{2}}x\right )}{\it \_C2}+{{\sl J}_{n+{\frac {1}{2}}}\left (\sqrt {-{a}^{2}}x\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[-((n*(1 + 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, (-I)*a*x]*C[1] + BesselY[1/2 + n, (-I)*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),'implicit')
Maple raw output
y(x) = x^(1/2)*(BesselY(n+1/2,(-a^2)^(1/2)*x)*_C2+BesselJ(n+1/2,(-a^2)^(1/2)*x)*_C1)