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 = 0.163389 (sec), leaf count = 79
\[\left \{\left \{y(x)\to -\frac {\sqrt {\frac {2}{\pi }} \sqrt {x} \left (\left (-a^2 c_2 x^2+3 a c_1 x+3 c_2\right ) \cos (a x)+\left (c_1 \left (a^2 x^2-3\right )+3 a c_2 x\right ) \sin (a x)\right )}{(a x)^{5/2}}\right \}\right \}\]
Maple ✓
cpu = 0.68 (sec), leaf count = 61
\[\left [y \left (x \right ) = \frac {\textit {\_C1} \left (\left (a^{2} x^{2}-3\right ) \cos \left (a x \right )-3 \sin \left (a x \right ) x a \right )}{x^{2}}+\frac {\textit {\_C2} \left (3 \cos \left (a x \right ) x a +\left (a^{2} x^{2}-3\right ) \sin \left (a x \right )\right )}{x^{2}}\right ]\] 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))
Maple raw output
[y(x) = _C1/x^2*((a^2*x^2-3)*cos(a*x)-3*sin(a*x)*x*a)+_C2/x^2*(3*cos(a*x)*x*a+(a^2*x^2-3)*sin(a*x))]