ODE
\[ 4 a x y'(x)-a (a+2) y(x)+4 \left (1-x^2\right ) y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.177357 (sec), leaf count = 84
\[\left \{\left \{y(x)\to \frac {\sqrt {1-x^2} \left (x^2-1\right )^{a/4} e^{-\frac {1}{2} \sqrt {(a+2)^2} \tanh ^{-1}(x)} \left (c_2 e^{\sqrt {(a+2)^2} \tanh ^{-1}(x)}+\sqrt {(a+2)^2} c_1\right )}{\sqrt {(a+2)^2}}\right \}\right \}\]
Maple ✓
cpu = 0.031 (sec), leaf count = 27
\[ \left \{ y \left ( x \right ) ={\it \_C1}\, \left ( 1+x \right ) ^{{\frac {a}{2}}+1}+{\it \_C2}\, \left ( -1+x \right ) ^{{\frac {a}{2}}+1} \right \} \] Mathematica raw input
DSolve[-(a*(2 + a)*y[x]) + 4*a*x*y'[x] + 4*(1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (Sqrt[1 - x^2]*(-1 + x^2)^(a/4)*(Sqrt[(2 + a)^2]*C[1] + E^(Sqrt[(2 + a)^2]*ArcTanh[x])*C[2]))/(Sqrt[(2 + a)^2]*E^((Sqrt[(2 + a)^2]*ArcTanh[x])/2))}}
Maple raw input
dsolve(4*(-x^2+1)*diff(diff(y(x),x),x)+4*a*x*diff(y(x),x)-a*(a+2)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*(1+x)^(1/2*a+1)+_C2*(-1+x)^(1/2*a+1)