ODE
\[ n (n+2) y(x)+\left (1-x^2\right ) y''(x)-3 x y'(x)=0 \] ODE Classification
[_Gegenbauer]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0222323 (sec), leaf count = 42
\[\left \{\left \{y(x)\to \frac {c_1 P_{n+\frac {1}{2}}^{\frac {1}{2}}(x)+c_2 Q_{n+\frac {1}{2}}^{\frac {1}{2}}(x)}{\sqrt [4]{x^2-1}}\right \}\right \}\]
Maple ✓
cpu = 0.108 (sec), leaf count = 68
\[ \left \{ y \left ( x \right ) ={1 \left ( {\it \_C1}\, \left ( -\sqrt {{x}^{2}-1}+x \right ) \left ( x+\sqrt {{x}^{2}-1} \right ) ^{-n-1}-{\it \_C2}\, \left ( x+\sqrt {{x}^{2}-1} \right ) ^{n} \right ) {\frac {1}{\sqrt {{x}^{2}-1}}} \left ( -\sqrt {{x}^{2}-1}+x \right ) ^{-1}} \right \} \] Mathematica raw input
DSolve[n*(2 + n)*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 + n, 1/2, x] + C[2]*LegendreQ[1/2 + n, 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)+n*(2+n)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = 1/(x^2-1)^(1/2)*(_C1*(-(x^2-1)^(1/2)+x)*(x+(x^2-1)^(1/2))^(-n-1)-_C2*(x+(x^2-1)^(1/2))^n)/(-(x^2-1)^(1/2)+x)