ODE
\[ (p-k) (k+p+1) y(x)+(k+1) (1-2 x) y'(x)+(1-x) x y''(x)=0 \] ODE Classification
[_Jacobi]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0269441 (sec), leaf count = 41
\[\left \{\left \{y(x)\to (-(x-1) x)^{-k/2} \left (c_1 P_p^k(2 x-1)+c_2 Q_p^k(2 x-1)\right )\right \}\right \}\]
Maple ✓
cpu = 0.063 (sec), leaf count = 45
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\mbox {$_2$F$_1$}(k-p,1+k+p;\,k+1;\,x)}+{\it \_C2}\,{x}^{-k}{\mbox {$_2$F$_1$}(-p,p+1;\,1-k;\,x)} \right \} \] Mathematica raw input
DSolve[(-k + p)*(1 + k + p)*y[x] + (1 + k)*(1 - 2*x)*y'[x] + (1 - x)*x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (C[1]*LegendreP[p, k, -1 + 2*x] + C[2]*LegendreQ[p, k, -1 + 2*x])/(-((-1 + x)*x))^(k/2)}}
Maple raw input
dsolve(x*(1-x)*diff(diff(y(x),x),x)+(k+1)*(1-2*x)*diff(y(x),x)+(p-k)*(1+k+p)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*hypergeom([k-p, 1+k+p],[k+1],x)+_C2*x^(-k)*hypergeom([-p, p+1],[1-k],x)