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 = 0.0166918 (sec), leaf count = 67
\[\left \{\left \{y(x)\to -\frac {\sqrt {\frac {2}{\pi }} a x^{3/2} \left (\left (c_1+i a c_2 x\right ) \sinh (a x)-\left (a c_1 x+i c_2\right ) \cosh (a x)\right )}{(-i a x)^{5/2}}\right \}\right \}\]
Maple ✓
cpu = 0.052 (sec), leaf count = 32
\[ \left \{ y \left ( x \right ) ={\frac {{\it \_C2}\, \left ( ax+1 \right ) {{\rm e}^{-ax}}+{{\rm e}^{ax}}{\it \_C1}\, \left ( ax-1 \right ) }{x}} \right \} \] 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),'implicit')
Maple raw output
y(x) = (_C2*(a*x+1)*exp(-a*x)+exp(a*x)*_C1*(a*x-1))/x