ODE
\[ -\left (\left (4 p^2+1\right ) y(x)\right )+4 \left (1-x^2\right ) y''(x)-8 x y'(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.163129 (sec), leaf count = 34
\[\left \{\left \{y(x)\to c_1 P_{i p-\frac {1}{2}}(x)+c_2 Q_{i p-\frac {1}{2}}(x)\right \}\right \}\]
Maple ✓
cpu = 0.273 (sec), leaf count = 25
\[\left [y \left (x \right ) = \textit {\_C1} \LegendreP \left (i p -\frac {1}{2}, x\right )+\textit {\_C2} \LegendreQ \left (i p -\frac {1}{2}, x\right )\right ]\] Mathematica raw input
DSolve[-((1 + 4*p^2)*y[x]) - 8*x*y'[x] + 4*(1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*LegendreP[-1/2 + I*p, x] + C[2]*LegendreQ[-1/2 + I*p, x]}}
Maple raw input
dsolve(4*(-x^2+1)*diff(diff(y(x),x),x)-8*x*diff(y(x),x)-(4*p^2+1)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*LegendreP(I*p-1/2,x)+_C2*LegendreQ(I*p-1/2,x)]