ODE
\[ -a y(x)+\left (1-x^2\right ) y''(x)-3 x y'(x)=0 \] ODE Classification
[_Gegenbauer]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0152825 (sec), leaf count = 58
\[\left \{\left \{y(x)\to \frac {c_1 P_{\sqrt {1-a}-\frac {1}{2}}^{\frac {1}{2}}(x)+c_2 Q_{\sqrt {1-a}-\frac {1}{2}}^{\frac {1}{2}}(x)}{\sqrt [4]{x^2-1}}\right \}\right \}\]
Maple ✓
cpu = 0.105 (sec), leaf count = 53
\[ \left \{ y \left ( x \right ) ={1 \left ( {\it \_C2}\, \left ( x+\sqrt {{x}^{2}-1} \right ) ^{-\sqrt {1-a}}+{\it \_C1}\, \left ( x+\sqrt {{x}^{2}-1} \right ) ^{\sqrt {1-a}} \right ) {\frac {1}{\sqrt {{x}^{2}-1}}}} \right \} \] Mathematica raw input
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 + Sqrt[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)