ODE
\[ a x^3 y(x)+\left (1-x^2\right ) x y''(x)-y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.176701 (sec), leaf count = 47
\[\left \{\left \{y(x)\to c_1 \cosh \left (\sqrt {a} \sqrt {x^2-1}\right )+i c_2 \sinh \left (\sqrt {a} \sqrt {x^2-1}\right )\right \}\right \}\]
Maple ✓
cpu = 0.086 (sec), leaf count = 45
\[\left [y \left (x \right ) = \textit {\_C1} \sinh \left (\frac {\left (x -1\right ) \left (x +1\right ) \sqrt {a}}{\sqrt {x^{2}-1}}\right )+\textit {\_C2} \cosh \left (\frac {\left (x -1\right ) \left (x +1\right ) \sqrt {a}}{\sqrt {x^{2}-1}}\right )\right ]\] Mathematica raw input
DSolve[a*x^3*y[x] - y'[x] + x*(1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Cosh[Sqrt[a]*Sqrt[-1 + x^2]] + I*C[2]*Sinh[Sqrt[a]*Sqrt[-1 + x^2]]}}
Maple raw input
dsolve(x*(-x^2+1)*diff(diff(y(x),x),x)-diff(y(x),x)+a*x^3*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*sinh((x-1)*(x+1)*a^(1/2)/(x^2-1)^(1/2))+_C2*cosh((x-1)*(x+1)*a^(1/2)/(x^2-1)^(1/2))]