4.32.4 \(-2 a x y'(x)+(1-a) a y(x)+\left (1-x^2\right ) y''(x)=0\)

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 = 0.200511 (sec), leaf count = 87

\[\left \{\left \{y(x)\to \frac {\sqrt {1-x^2} \left (x^2-1\right )^{-a/2} e^{-\sqrt {(a-1)^2} \tanh ^{-1}(x)} \left (c_2 e^{2 \sqrt {(a-1)^2} \tanh ^{-1}(x)}+2 \sqrt {(a-1)^2} c_1\right )}{2 \sqrt {(a-1)^2}}\right \}\right \}\]

Maple ✓
cpu = 0.028 (sec), leaf count = 27

\[ \left \{ y \left ( x \right ) ={\it \_C1}\, \left ( 1+x \right ) ^{1-a}+{\it \_C2}\, \left ( -1+x \right ) ^{1-a} \right \} \] 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]*C[1] + E^(2*Sqrt[(-1 + a)^2]*ArcTan
h[x])*C[2]))/(2*Sqrt[(-1 + a)^2]*E^(Sqrt[(-1 + a)^2]*ArcTanh[x])*(-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),'implicit')

Maple raw output

y(x) = _C1*(1+x)^(1-a)+_C2*(-1+x)^(1-a)