ODE
\[ p (p+1) y(x)+\left (1-x^2\right ) y''(x)-2 x y'(x)=0 \] ODE Classification
[_Gegenbauer]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0120584 (sec), leaf count = 18
\[\left \{\left \{y(x)\to c_1 P_p(x)+c_2 Q_p(x)\right \}\right \}\]
Maple ✓
cpu = 0.049 (sec), leaf count = 15
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\it LegendreP} \left ( p,x \right ) +{\it \_C2}\,{\it LegendreQ} \left ( p,x \right ) \right \} \] Mathematica raw input
DSolve[p*(1 + p)*y[x] - 2*x*y'[x] + (1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*LegendreP[p, x] + C[2]*LegendreQ[p, x]}}
Maple raw input
dsolve((-x^2+1)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+p*(p+1)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*LegendreP(p,x)+_C2*LegendreQ(p,x)