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 = 0.319589 (sec), leaf count = 125
\[\left \{\left \{y(x)\to \frac {\sqrt {1-x^2} (1-x)^{-\frac {1}{2} \sqrt {(a+2)^2}} \left (x^2-1\right )^{a/4} \left (\sqrt {(a+2)^2} c_1 (1-x)^{\frac {1}{2} \sqrt {(a+2)^2}}+c_2 (x+1)^{\frac {1}{2} \sqrt {(a+2)^2}}\right ) e^{-\frac {1}{2} \sqrt {(a+2)^2} \tanh ^{-1}(x)}}{\sqrt {(a+2)^2}}\right \}\right \}\]
Maple ✓
cpu = 0.092 (sec), leaf count = 27
\[\left [y \left (x \right ) = \textit {\_C1} \left (x +1\right )^{\frac {a}{2}+1}+\textit {\_C2} \left (x -1\right )^{\frac {a}{2}+1}\right ]\] 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]*(1 - x)^(Sqrt[(2 + a)^2]/2)*C[1] + (1 + x)^(Sqrt[(2 + a)^2]/2)*C[2]))/(Sqrt[(2 + a)^2]*E^((Sqrt[(2 + a)^2]*ArcTanh[x])/2)*(1 - x)^(Sqrt[(2 + a)^2]/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))
Maple raw output
[y(x) = _C1*(x+1)^(1/2*a+1)+_C2*(x-1)^(1/2*a+1)]