ODE
\[ x^2 y''(x)-\left (a^2 x^2+2\right ) y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.165471 (sec), leaf count = 67
\[\left \{\left \{y(x)\to -\frac {\sqrt {\frac {2}{\pi }} a x^{3/2} ((i a c_2 x+c_1) \sinh (a x)-(a c_1 x+i c_2) \cosh (a x))}{(-i a x)^{5/2}}\right \}\right \}\]
Maple ✓
cpu = 0.077 (sec), leaf count = 34
\[\left [y \left (x \right ) = \frac {\textit {\_C1} \,{\mathrm e}^{a x} \left (a x -1\right )}{x}+\frac {\textit {\_C2} \,{\mathrm e}^{-a x} \left (a x +1\right )}{x}\right ]\] Mathematica raw input
DSolve[-((2 + a^2*x^2)*y[x]) + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -((a*Sqrt[2/Pi]*x^(3/2)*(-((a*x*C[1] + I*C[2])*Cosh[a*x]) + (C[1] + I*a*x*C[2])*Sinh[a*x]))/((-I)*a*x)^(5/2))}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)-(a^2*x^2+2)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1/x*exp(a*x)*(a*x-1)+_C2/x*exp(-a*x)*(a*x+1)]