4.31.50 \(n y(x) (a+b+n+1)+(-x (a+b+2)-a+b) y'(x)+\left (1-x^2\right ) y''(x)=0\)

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 = 0.29157 (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 = 0.437 (sec), leaf count = 61

\[\left [y \left (x \right ) = \textit {\_C1} \hypergeom \left (\left [-n , 1+a +b +n \right ], \left [b +1\right ], \frac {1}{2}+\frac {x}{2}\right )+\textit {\_C2} \left (\frac {1}{2}+\frac {x}{2}\right )^{-b} \hypergeom \left (\left [-n -b , 1+a +n \right ], \left [1-b \right ], \frac {1}{2}+\frac {x}{2}\right )\right ]\] 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))

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)*hyperg
eom([-n-b, 1+a+n],[1-b],1/2+1/2*x)]