ODE
\[ a y(x)+2 (1-x) x y''(x)-(1-2 x) y'(x)=0 \] ODE Classification
[_Jacobi]
Book solution method
TO DO
Mathematica ✓
cpu = 0.169819 (sec), leaf count = 75
\[\left \{\left \{y(x)\to (-((x-1) x))^{3/4} \left (c_1 P_{\frac {\sqrt {a+2}}{\sqrt {2}}-\frac {1}{2}}^{\frac {3}{2}}(2 x-1)+c_2 Q_{\frac {\sqrt {a+2}}{\sqrt {2}}-\frac {1}{2}}^{\frac {3}{2}}(2 x-1)\right )\right \}\right \}\]
Maple ✓
cpu = 0.339 (sec), leaf count = 66
\[\left [y \left (x \right ) = \textit {\_C1} \hypergeom \left (\left [-1-\frac {\sqrt {2 a +4}}{2}, -1+\frac {\sqrt {2 a +4}}{2}\right ], \left [-\frac {1}{2}\right ], x\right )+\textit {\_C2} \,x^{\frac {3}{2}} \hypergeom \left (\left [\frac {1}{2}-\frac {\sqrt {2 a +4}}{2}, \frac {1}{2}+\frac {\sqrt {2 a +4}}{2}\right ], \left [\frac {5}{2}\right ], x\right )\right ]\] Mathematica raw input
DSolve[a*y[x] - (1 - 2*x)*y'[x] + 2*(1 - x)*x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (-((-1 + x)*x))^(3/4)*(C[1]*LegendreP[-1/2 + Sqrt[2 + a]/Sqrt[2], 3/2, -1 + 2*x] + C[2]*LegendreQ[-1/2 + Sqrt[2 + a]/Sqrt[2], 3/2, -1 + 2*x])}}
Maple raw input
dsolve(2*x*(1-x)*diff(diff(y(x),x),x)-(1-2*x)*diff(y(x),x)+a*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*hypergeom([-1-1/2*(2*a+4)^(1/2), -1+1/2*(2*a+4)^(1/2)],[-1/2],x)+_C2*x^(3/2)*hypergeom([1/2-1/2*(2*a+4)^(1/2), 1/2+1/2*(2*a+4)^(1/2)],[5/2],x)]