4.33.39 $4ax{y}^{\prime }(x)-a(a+2)y(x)+4(1-x^2)y^{″}(x)=0$

ODE
$$4ax{y}^{\prime }(x)-a(a+2)y(x)+4(1-x^2)y^{″}(x)=0$$ ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.177357 (sec), leaf count = 84

$$\left\{\{y(x)\to \frac{\sqrt{1-x^2}{(x^2-1)}^{a/4}{e}^{-\frac{1}{2}\sqrt{(a+2{)}^2}{\tanh }^{-1}(x)}(c_2{e}^{\sqrt{(a+2{)}^2}{\tanh }^{-1}(x)}+\sqrt{(a+2{)}^2}{c}_1)}{\sqrt{(a+2{)}^2}}\}\right\}$$

Maple ✓
cpu = 0.031 (sec), leaf count = 27

$$\{y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}{(1+x)}^{\frac{a}{2}+1}+\mathit{\_}\mathit{C}\mathit{2}{(-1+x)}^{\frac{a}{2}+1}\}$$ 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)