ODE
\[ n y(x) (a+b+n+1)+(-x (a+b+2)-a+b) y'(x)+\left (1-x^2\right ) y''(x)=0 \] ODE Classification
[_Gegenbauer]
Book solution method
TO DO
Mathematica ✓
cpu = 0.144938 (sec), leaf count = 69
\[\left \{\left \{y(x)\to 2^a c_2 (x-1)^{-a} \, _2F_1\left (-a-n,b+n+1;1-a;\frac {1-x}{2}\right )+c_1 \, _2F_1\left (-n,a+b+n+1;a+1;\frac {1-x}{2}\right )\right \}\right \}\]
Maple ✓
cpu = 0.056 (sec), leaf count = 31
\[ \left \{ y \left ( x \right ) = \left ( {x}^{2}-1 \right ) ^{-{\frac {k}{2}}} \left ( {\it LegendreQ} \left ( k+p,k,x \right ) {\it \_C2}+{\it LegendreP} \left ( k+p,k,x \right ) {\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[n*(1 + a + b + n)*y[x] + (-a + b - (2 + a + b)*x)*y'[x] + (1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (2^a*C[2]*Hypergeometric2F1[-a - n, 1 + b + n, 1 - a, (1 - x)/2])/(-1 + x)^a + C[1]*Hypergeometric2F1[-n, 1 + a + b + n, 1 + a, (1 - x)/2]}}
Maple raw input
dsolve((-x^2+1)*diff(diff(y(x),x),x)-2*(k+1)*x*diff(y(x),x)+p*(1+p+2*k)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (x^2-1)^(-1/2*k)*(LegendreQ(k+p,k,x)*_C2+LegendreP(k+p,k,x)*_C1)