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 = 0.0286874 (sec), leaf count = 32
\[\left \{\left \{y(x)\to \left (x^2-1\right )^{-k/2} \left (c_1 P_p^k(x)+c_2 Q_p^k(x)\right )\right \}\right \}\]
Maple ✓
cpu = 0.057 (sec), leaf count = 27
\[ \left \{ y \left ( x \right ) = \left ( {x}^{2}-1 \right ) ^{-{\frac {k}{2}}} \left ( {\it LegendreP} \left ( p,k,x \right ) {\it \_C1}+{\it LegendreQ} \left ( p,k,x \right ) {\it \_C2} \right ) \right \} \] 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),'implicit')
Maple raw output
y(x) = (x^2-1)^(-1/2*k)*(LegendreP(p,k,x)*_C1+LegendreQ(p,k,x)*_C2)