ODE
\[ a x y'(x)+b y(x)+\left (1-x^2\right ) y''(x)=0 \] ODE Classification
[_Gegenbauer]
Book solution method
TO DO
Mathematica ✓
cpu = 0.182033 (sec), leaf count = 86
\[\left \{\left \{y(x)\to \left (x^2-1\right )^{\frac {a+2}{4}} \left (c_1 P_{\frac {1}{2} \left (\sqrt {a^2+2 a+4 b+1}-1\right )}^{\frac {a+2}{2}}(x)+c_2 Q_{\frac {1}{2} \left (\sqrt {a^2+2 a+4 b+1}-1\right )}^{\frac {a+2}{2}}(x)\right )\right \}\right \}\]
Maple ✓
cpu = 0.435 (sec), leaf count = 79
\[\left [y \left (x \right ) = \textit {\_C1} \left (x^{2}-1\right )^{\frac {a}{4}+\frac {1}{2}} \LegendreP \left (\frac {\sqrt {a^{2}+2 a +4 b +1}}{2}-\frac {1}{2}, \frac {a}{2}+1, x\right )+\textit {\_C2} \left (x^{2}-1\right )^{\frac {a}{4}+\frac {1}{2}} \LegendreQ \left (\frac {\sqrt {a^{2}+2 a +4 b +1}}{2}-\frac {1}{2}, \frac {a}{2}+1, x\right )\right ]\] Mathematica raw input
DSolve[b*y[x] + a*x*y'[x] + (1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (-1 + x^2)^((2 + a)/4)*(C[1]*LegendreP[(-1 + Sqrt[1 + 2*a + a^2 + 4*b])/2, (2 + a)/2, x] + C[2]*LegendreQ[(-1 + Sqrt[1 + 2*a + a^2 + 4*b])/2, (2 + a)/2, x])}}
Maple raw input
dsolve((-x^2+1)*diff(diff(y(x),x),x)+a*x*diff(y(x),x)+b*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*(x^2-1)^(1/4*a+1/2)*LegendreP(1/2*(a^2+2*a+4*b+1)^(1/2)-1/2,1/2*a+1,x)+_C2*(x^2-1)^(1/4*a+1/2)*LegendreQ(1/2*(a^2+2*a+4*b+1)^(1/2)-1/2,1/2*a+1,x)]