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 = 0.023909 (sec), leaf count = 38
\[\left \{\left \{y(x)\to c_2 \sin \left (\log \left (\sqrt {x^2-1}+x\right )\right )+c_1 \cos \left (\log \left (\sqrt {x^2-1}+x\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.012 (sec), leaf count = 37
\[ \left \{ y \left ( x \right ) ={1 \left ( {\it \_C1}\, \left ( \left ( x+\sqrt {{x}^{2}-1} \right ) ^{i} \right ) ^{2}+{\it \_C2} \right ) \left ( \left ( x+\sqrt {{x}^{2}-1} \right ) ^{i} \right ) ^{-1}} \right \} \] 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[x + Sqrt[-1 + x^2]]] + C[2]*Sin[Log[x + Sqrt[-1 + x^2]]]}}
Maple raw input
dsolve((-x^2+1)*diff(diff(y(x),x),x)-x*diff(y(x),x)-y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (_C1*((x+(x^2-1)^(1/2))^I)^2+_C2)/((x+(x^2-1)^(1/2))^I)