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 = 0.0281031 (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 = 0.021 (sec), leaf count = 45
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,\sinh \left ( { \left ( -1+x \right ) \left ( 1+x \right ) \sqrt {a}{\frac {1}{\sqrt {{x}^{2}-1}}}} \right ) +{\it \_C2}\,\cosh \left ( { \left ( -1+x \right ) \left ( 1+x \right ) \sqrt {a}{\frac {1}{\sqrt {{x}^{2}-1}}}} \right ) \right \} \] 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),'implicit')
Maple raw output
y(x) = _C1*sinh((-1+x)*(1+x)/(x^2-1)^(1/2)*a^(1/2))+_C2*cosh((-1+x)*(1+x)/(x^2-1)^(1/2)*a^(1/2))