ODE
\[ x^2 y''(x)-\left (6-a^2 x^2\right ) y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0141424 (sec), leaf count = 79
\[\left \{\left \{y(x)\to -\frac {\sqrt {\frac {2}{\pi }} \sqrt {x} \left (\left (c_1 \left (a^2 x^2-3\right )+3 a c_2 x\right ) \sin (a x)+\left (-a^2 c_2 x^2+3 a c_1 x+3 c_2\right ) \cos (a x)\right )}{(a x)^{5/2}}\right \}\right \}\]
Maple ✓
cpu = 0.244 (sec), leaf count = 53
\[ \left \{ y \left ( x \right ) ={\frac { \left ( {\it \_C1}\,{a}^{2}{x}^{2}+3\,{\it \_C2}\,ax-3\,{\it \_C1} \right ) \cos \left ( ax \right ) +\sin \left ( ax \right ) \left ( {\it \_C2}\,{a}^{2}{x}^{2}-3\,{\it \_C1}\,ax-3\,{\it \_C2} \right ) }{{x}^{2}}} \right \} \] Mathematica raw input
DSolve[-((6 - a^2*x^2)*y[x]) + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -((Sqrt[2/Pi]*Sqrt[x]*((3*a*x*C[1] + 3*C[2] - a^2*x^2*C[2])*Cos[a*x] + ((-3 + a^2*x^2)*C[1] + 3*a*x*C[2])*Sin[a*x]))/(a*x)^(5/2))}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)-(-a^2*x^2+6)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = ((_C1*a^2*x^2+3*_C2*a*x-3*_C1)*cos(a*x)+sin(a*x)*(_C2*a^2*x^2-3*_C1*a*x-3*_C2))/x^2