ODE
\[ -y(x) \left (4 k^2+\left (1-4 p^2\right ) \left (1-x^2\right )\right )+4 \left (1-x^2\right )^2 y''(x)-8 x \left (1-x^2\right ) y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.02868 (sec), leaf count = 28
\[\left \{\left \{y(x)\to c_1 P_{p-\frac {1}{2}}^k(x)+c_2 Q_{p-\frac {1}{2}}^k(x)\right \}\right \}\]
Maple ✓
cpu = 0.054 (sec), leaf count = 21
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\it LegendreP} \left ( p-{\frac {1}{2}},k,x \right ) +{\it \_C2}\,{\it LegendreQ} \left ( p-{\frac {1}{2}},k,x \right ) \right \} \] Mathematica raw input
DSolve[-((4*k^2 + (1 - 4*p^2)*(1 - x^2))*y[x]) - 8*x*(1 - x^2)*y'[x] + 4*(1 - x^2)^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*LegendreP[-1/2 + p, k, x] + C[2]*LegendreQ[-1/2 + p, k, x]}}
Maple raw input
dsolve(4*(-x^2+1)^2*diff(diff(y(x),x),x)-8*x*(-x^2+1)*diff(y(x),x)-(4*k^2+(-4*p^2+1)*(-x^2+1))*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*LegendreP(p-1/2,k,x)+_C2*LegendreQ(p-1/2,k,x)