ODE
\[ \left (1-x^2\right ) y''(x)+x y'(x)-y(x)=0 \] ODE Classification
[_Gegenbauer]
Book solution method
TO DO
Mathematica ✓
cpu = 0.188916 (sec), leaf count = 50
\[\left \{\left \{y(x)\to c_1 x-\frac {c_2 \sqrt {x^2-1} \left (\sqrt {1-x^2}+x \sin ^{-1}(x)\right )}{\sqrt {1-x^2}}\right \}\right \}\]
Maple ✓
cpu = 0.112 (sec), leaf count = 32
\[\left [y \left (x \right ) = \textit {\_C1} x +\textit {\_C2} \left (\ln \left (x +\sqrt {x^{2}-1}\right ) x -\sqrt {x^{2}-1}\right )\right ]\] Mathematica raw input
DSolve[-y[x] + x*y'[x] + (1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> x*C[1] - (Sqrt[-1 + x^2]*(Sqrt[1 - x^2] + x*ArcSin[x])*C[2])/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))
Maple raw output
[y(x) = _C1*x+_C2*(ln(x+(x^2-1)^(1/2))*x-(x^2-1)^(1/2))]