ODE
\[ n y(x) (a+b+n+1)+(-x (a+b+2)-a+b) y'(x)+\left (1-x^2\right ) y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.248012 (sec), leaf count = 69
\[\left \{\left \{y(x)\to 2^a c_2 (x-1)^{-a} \, _2F_1\left (-a-n,b+n+1;1-a;\frac {1-x}{2}\right )+c_1 \, _2F_1\left (-n,a+b+n+1;a+1;\frac {1-x}{2}\right )\right \}\right \}\]
Maple ✓
cpu = 0.081 (sec), leaf count = 61
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\mbox {$_2$F$_1$}(-n,1+a+b+n;\,b+1;\,{\frac {1}{2}}+{\frac {x}{2}})}+{\it \_C2}\, \left ( {\frac {1}{2}}+{\frac {x}{2}} \right ) ^{-b}{\mbox {$_2$F$_1$}(-n-b,1+a+n;\,1-b;\,{\frac {1}{2}}+{\frac {x}{2}})} \right \} \] Mathematica raw input
DSolve[n*(1 + a + b + n)*y[x] + (-a + b - (2 + a + b)*x)*y'[x] + (1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (2^a*C[2]*Hypergeometric2F1[-a - n, 1 + b + n, 1 - a, (1 - x)/2])/(-1 + x)^a + C[1]*Hypergeometric2F1[-n, 1 + a + b + n, 1 + a, (1 - x)/2]}}
Maple raw input
dsolve((-x^2+1)*diff(diff(y(x),x),x)+(b-a-(2+a+b)*x)*diff(y(x),x)+n*(1+a+b+n)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*hypergeom([-n, 1+a+b+n],[b+1],1/2+1/2*x)+_C2*(1/2+1/2*x)^(-b)*hypergeom([-n-b, 1+a+n],[1-b],1/2+1/2*x)