ODE
\[ -2 a x y'(x)+(1-a) a y(x)+\left (1-x^2\right ) y''(x)=0 \] ODE Classification
[_Gegenbauer]
Book solution method
TO DO
Mathematica ✓
cpu = 0.331291 (sec), leaf count = 117
\[\left \{\left \{y(x)\to \frac {\sqrt {1-x^2} (1-x)^{-\sqrt {(a-1)^2}} \left (x^2-1\right )^{-a/2} \left (2 \sqrt {(a-1)^2} c_1 (1-x)^{\sqrt {(a-1)^2}}+c_2 (x+1)^{\sqrt {(a-1)^2}}\right ) e^{-\sqrt {(a-1)^2} \tanh ^{-1}(x)}}{2 \sqrt {(a-1)^2}}\right \}\right \}\]
Maple ✓
cpu = 0.086 (sec), leaf count = 27
\[[y \left (x \right ) = \textit {\_C1} \left (x +1\right )^{1-a}+\textit {\_C2} \left (x -1\right )^{1-a}]\] Mathematica raw input
DSolve[(1 - a)*a*y[x] - 2*a*x*y'[x] + (1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (Sqrt[1 - x^2]*(2*Sqrt[(-1 + a)^2]*(1 - x)^Sqrt[(-1 + a)^2]*C[1] + (1 + x)^Sqrt[(-1 + a)^2]*C[2]))/(2*Sqrt[(-1 + a)^2]*E^(Sqrt[(-1 + a)^2]*ArcTanh[x])*(1 - x)^Sqrt[(-1 + a)^2]*(-1 + x^2)^(a/2))}}
Maple raw input
dsolve((-x^2+1)*diff(diff(y(x),x),x)-2*a*x*diff(y(x),x)+a*(1-a)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*(x+1)^(1-a)+_C2*(x-1)^(1-a)]