[_Gegenbauer]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0152825 (sec), leaf count = 58
Maple ✓
cpu = 0.105 (sec), leaf count = 53
DSolve[-(a*y[x]) - 3*x*y'[x] + (1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (C[1]*LegendreP[-1/2 + Sqrt[1 - a], 1/2, x] + C[2]*LegendreQ[-1/2 + Sq
rt[1 - a], 1/2, x])/(-1 + x^2)^(1/4)}}
Maple raw input
dsolve((-x^2+1)*diff(diff(y(x),x),x)-3*x*diff(y(x),x)-a*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (_C2*(x+(x^2-1)^(1/2))^(-(1-a)^(1/2))+_C1*(x+(x^2-1)^(1/2))^((1-a)^(1/2))
)/(x^2-1)^(1/2)