ODE
\[ n (n+2) y(x)+\left (1-x^2\right ) y''(x)-3 x y'(x)=0 \] ODE Classification
[_Gegenbauer]
Book solution method
TO DO
Mathematica ✓
cpu = 0.175994 (sec), leaf count = 42
\[\left \{\left \{y(x)\to \frac {c_1 P_{n+\frac {1}{2}}^{\frac {1}{2}}(x)+c_2 Q_{n+\frac {1}{2}}^{\frac {1}{2}}(x)}{\sqrt [4]{x^2-1}}\right \}\right \}\]
Maple ✓
cpu = 0.325 (sec), leaf count = 62
\[\left [y \left (x \right ) = \frac {\textit {\_C1} \left (x +\sqrt {x^{2}-1}\right )^{-1-n}}{\sqrt {x^{2}-1}}+\frac {\textit {\_C2} \left (x +\sqrt {x^{2}-1}\right )^{n}}{\left (\sqrt {x^{2}-1}-x \right ) \sqrt {x^{2}-1}}\right ]\] Mathematica raw input
DSolve[n*(2 + n)*y[x] - 3*x*y'[x] + (1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (C[1]*LegendreP[1/2 + n, 1/2, x] + C[2]*LegendreQ[1/2 + n, 1/2, x])/(-1 + x^2)^(1/4)}}
Maple raw input
dsolve((-x^2+1)*diff(diff(y(x),x),x)-3*x*diff(y(x),x)+n*(2+n)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*(x+(x^2-1)^(1/2))^(-1-n)/(x^2-1)^(1/2)+_C2/((x^2-1)^(1/2)-x)*(x+(x^2-1)^(1/2))^n/(x^2-1)^(1/2)]