ODE
\[ \left (1-x^2\right ) y''(x)-x y'(x)-y(x)=0 \] ODE Classification
[_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.194694 (sec), leaf count = 86
\[\left \{\left \{y(x)\to c_1 \cos \left (\frac {1}{2} \left (\log \left (\frac {x}{\sqrt {x^2-1}}+1\right )-\log \left (1-\frac {x}{\sqrt {x^2-1}}\right )\right )\right )+c_2 \sin \left (\frac {1}{2} \left (\log \left (\frac {x}{\sqrt {x^2-1}}+1\right )-\log \left (1-\frac {x}{\sqrt {x^2-1}}\right )\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.109 (sec), leaf count = 35
\[\left [y \left (x \right ) = \textit {\_C1} \left (x +\sqrt {x^{2}-1}\right )^{i}+\textit {\_C2} \left (x +\sqrt {x^{2}-1}\right )^{-i}\right ]\] Mathematica raw input
DSolve[-y[x] - x*y'[x] + (1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Cos[(-Log[1 - x/Sqrt[-1 + x^2]] + Log[1 + x/Sqrt[-1 + x^2]])/2] + C[2]*Sin[(-Log[1 - x/Sqrt[-1 + x^2]] + Log[1 + x/Sqrt[-1 + x^2]])/2]}}
Maple raw input
dsolve((-x^2+1)*diff(diff(y(x),x),x)-x*diff(y(x),x)-y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*(x+(x^2-1)^(1/2))^I+_C2/((x+(x^2-1)^(1/2))^I)]