ODE
\[ p (p+1) y(x)+(1-x) x y''(x)+(1-2 x) y'(x)=0 \] ODE Classification
[_Jacobi]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0415777 (sec), leaf count = 26
\[\left \{\left \{y(x)\to c_1 P_p(2 x-1)+c_2 Q_p(2 x-1)\right \}\right \}\]
Maple ✓
cpu = 0.125 (sec), leaf count = 51
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\mbox {$_2$F$_1$}(-p,-p;\,-2\,p;\,{x}^{-1})}{x}^{p}+{\it \_C2}\,{\mbox {$_2$F$_1$}(p+1,p+1;\,2+2\,p;\,{x}^{-1})}{x}^{-p-1} \right \} \] Mathematica raw input
DSolve[p*(1 + p)*y[x] + (1 - 2*x)*y'[x] + (1 - x)*x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*LegendreP[p, -1 + 2*x] + C[2]*LegendreQ[p, -1 + 2*x]}}
Maple raw input
dsolve(x*(1-x)*diff(diff(y(x),x),x)+(1-2*x)*diff(y(x),x)+p*(p+1)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*hypergeom([-p, -p],[-2*p],1/x)*x^p+_C2*hypergeom([p+1, p+1],[2+2*p],1/x)*x^(-p-1)