ODE
\[ p (2 k+p) y(x)-(2 k+1) x y'(x)+\left (1-x^2\right ) y''(x)=0 \] ODE Classification
[_Gegenbauer]
Book solution method
TO DO
Mathematica ✓
cpu = 0.190124 (sec), leaf count = 54
\[\left \{\left \{y(x)\to \left (x^2-1\right )^{\frac {1}{4}-\frac {k}{2}} \left (c_1 P_{k+p-\frac {1}{2}}^{k-\frac {1}{2}}(x)+c_2 Q_{k+p-\frac {1}{2}}^{k-\frac {1}{2}}(x)\right )\right \}\right \}\]
Maple ✓
cpu = 0.443 (sec), leaf count = 49
\[\left [y \left (x \right ) = \textit {\_C1} \left (x^{2}-1\right )^{\frac {1}{4}-\frac {k}{2}} \LegendreP \left (k +p -\frac {1}{2}, -\frac {1}{2}+k , x\right )+\textit {\_C2} \left (x^{2}-1\right )^{\frac {1}{4}-\frac {k}{2}} \LegendreQ \left (k +p -\frac {1}{2}, -\frac {1}{2}+k , x\right )\right ]\] Mathematica raw input
DSolve[p*(2*k + p)*y[x] - (1 + 2*k)*x*y'[x] + (1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (-1 + x^2)^(1/4 - k/2)*(C[1]*LegendreP[-1/2 + k + p, -1/2 + k, x] + C[2]*LegendreQ[-1/2 + k + p, -1/2 + k, x])}}
Maple raw input
dsolve((-x^2+1)*diff(diff(y(x),x),x)-(1+2*k)*x*diff(y(x),x)+p*(p+2*k)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*(x^2-1)^(1/4-1/2*k)*LegendreP(k+p-1/2,-1/2+k,x)+_C2*(x^2-1)^(1/4-1/2*k)*LegendreQ(k+p-1/2,-1/2+k,x)]