ODE
\[ -((k-p) (k+p+1) y(x))-2 (k+1) x y'(x)+\left (1-x^2\right ) y''(x)=0 \] ODE Classification
[_Gegenbauer]
Book solution method
TO DO
Mathematica ✓
cpu = 0.173383 (sec), leaf count = 32
\[\left \{\left \{y(x)\to \left (x^2-1\right )^{-k/2} (c_1 P_p^k(x)+c_2 Q_p^k(x))\right \}\right \}\]
Maple ✓
cpu = 0.34 (sec), leaf count = 35
\[\left [y \left (x \right ) = \textit {\_C1} \left (x^{2}-1\right )^{-\frac {k}{2}} \LegendreP \left (p , k , x\right )+\textit {\_C2} \left (x^{2}-1\right )^{-\frac {k}{2}} \LegendreQ \left (p , k , x\right )\right ]\] Mathematica raw input
DSolve[-((k - p)*(1 + k + p)*y[x]) - 2*(1 + k)*x*y'[x] + (1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (C[1]*LegendreP[p, k, x] + C[2]*LegendreQ[p, k, x])/(-1 + x^2)^(k/2)}}
Maple raw input
dsolve((-x^2+1)*diff(diff(y(x),x),x)-2*(k+1)*x*diff(y(x),x)-(k-p)*(1+k+p)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*(x^2-1)^(-1/2*k)*LegendreP(p,k,x)+_C2*(x^2-1)^(-1/2*k)*LegendreQ(p,k,x)]