ODE
\[ (c-(a+1) x) y'(x)+n (a+n) y(x)+(1-x) x y''(x)=0 \] ODE Classification
[_Jacobi]
Book solution method
TO DO
Mathematica ✓
cpu = 0.281872 (sec), leaf count = 57
\[\left \{\left \{y(x)\to (-1)^{1-c} c_2 x^{1-c} \, _2F_1(-c-n+1,a-c+n+1;2-c;x)+c_1 \, _2F_1(-n,a+n;c;x)\right \}\right \}\]
Maple ✓
cpu = 0.437 (sec), leaf count = 51
\[[y \left (x \right ) = \textit {\_C1} \hypergeom \left (\left [-n , n +a \right ], \left [c \right ], x\right )+\textit {\_C2} \,x^{-c +1} \hypergeom \left (\left [-n -c +1, n +a -c +1\right ], \left [-c +2\right ], x\right )]\] Mathematica raw input
DSolve[n*(a + n)*y[x] + (c - (1 + a)*x)*y'[x] + (1 - x)*x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (-1)^(1 - c)*x^(1 - c)*C[2]*Hypergeometric2F1[1 - c - n, 1 + a - c + n, 2 - c, x] + C[1]*Hypergeometric2F1[-n, a + n, c, x]}}
Maple raw input
dsolve(x*(1-x)*diff(diff(y(x),x),x)+(c-(1+a)*x)*diff(y(x),x)+n*(n+a)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*hypergeom([-n, n+a],[c],x)+_C2*x^(-c+1)*hypergeom([-n-c+1, n+a-c+1],[-c+2],x)]