4.32.34 \(y'(x) (c-x (a+b+1))-a b y(x)+(1-x) x y''(x)=0\)

ODE
\[ y'(x) (c-x (a+b+1))-a b y(x)+(1-x) x y''(x)=0 \] ODE Classification

[_Jacobi]

Book solution method
TO DO

Mathematica ✓
cpu = 0.149333 (sec), leaf count = 50

\[\left \{\left \{y(x)\to (-1)^{1-c} c_2 x^{1-c} \, _2F_1(a-c+1,b-c+1;2-c;x)+c_1 \, _2F_1(a,b;c;x)\right \}\right \}\]

Maple ✓
cpu = 0.059 (sec), leaf count = 44

\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\mbox {$_2$F$_1$}(a,b;\,c;\,x)}+{\it \_C2}\,{x}^{-c+1}{\mbox {$_2$F$_1$}(b-c+1,a-c+1;\,-c+2;\,x)} \right \} \] Mathematica raw input

DSolve[-(a*b*y[x]) + (c - (1 + a + b)*x)*y'[x] + (1 - x)*x*y''[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> C[1]*Hypergeometric2F1[a, b, c, x] + (-1)^(1 - c)*x^(1 - c)*C[2]*Hyper
geometric2F1[1 + a - c, 1 + b - c, 2 - c, x]}}

Maple raw input

dsolve(x*(1-x)*diff(diff(y(x),x),x)+(c-(1+a+b)*x)*diff(y(x),x)-a*b*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = _C1*hypergeom([a, b],[c],x)+_C2*x^(-c+1)*hypergeom([b-c+1, a-c+1],[-c+2],
x)