ODE
\[ (a+b x) y'(x)+c y(x)+(1-x) x y''(x)=0 \] ODE Classification
[_Jacobi]
Book solution method
TO DO
Mathematica ✓
cpu = 0.138508 (sec), leaf count = 142
\[\left \{\left \{y(x)\to (-1)^{1-a} c_2 x^{1-a} \, _2F_1\left (\frac {1}{2} \left (-2 a-b-\sqrt {b^2+2 b+4 c+1}+1\right ),\frac {1}{2} \left (-2 a-b+\sqrt {b^2+2 b+4 c+1}+1\right );2-a;x\right )+c_1 \, _2F_1\left (\frac {1}{2} \left (-b-\sqrt {b^2+2 b+4 c+1}-1\right ),\frac {1}{2} \left (-b+\sqrt {b^2+2 b+4 c+1}-1\right );a;x\right )\right \}\right \}\]
Maple ✓
cpu = 0.067 (sec), leaf count = 116
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\mbox {$_2$F$_1$}(-{\frac {1}{2}}-{\frac {b}{2}}-{\frac {1}{2}\sqrt {{b}^{2}+2\,b+4\,c+1}},-{\frac {1}{2}}-{\frac {b}{2}}+{\frac {1}{2}\sqrt {{b}^{2}+2\,b+4\,c+1}};\,a;\,x)}+{\it \_C2}\,{x}^{1-a}{\mbox {$_2$F$_1$}({\frac {1}{2}}-{\frac {b}{2}}-{\frac {1}{2}\sqrt {{b}^{2}+2\,b+4\,c+1}}-a,{\frac {1}{2}}-{\frac {b}{2}}+{\frac {1}{2}\sqrt {{b}^{2}+2\,b+4\,c+1}}-a;\,2-a;\,x)} \right \} \] Mathematica raw input
DSolve[c*y[x] + (a + b*x)*y'[x] + (1 - x)*x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Hypergeometric2F1[(-1 - b - Sqrt[1 + 2*b + b^2 + 4*c])/2, (-1 - b + Sqrt[1 + 2*b + b^2 + 4*c])/2, a, x] + (-1)^(1 - a)*x^(1 - a)*C[2]*Hypergeometric2F1[(1 - 2*a - b - Sqrt[1 + 2*b + b^2 + 4*c])/2, (1 - 2*a - b + Sqrt[1 + 2*b + b^2 + 4*c])/2, 2 - a, x]}}
Maple raw input
dsolve(x*(1-x)*diff(diff(y(x),x),x)+(b*x+a)*diff(y(x),x)+c*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*hypergeom([-1/2-1/2*b-1/2*(b^2+2*b+4*c+1)^(1/2), -1/2-1/2*b+1/2*(b^2+2*b+4*c+1)^(1/2)],[a],x)+_C2*x^(1-a)*hypergeom([1/2-1/2*b-1/2*(b^2+2*b+4*c+1)^(1/2)-a, 1/2-1/2*b+1/2*(b^2+2*b+4*c+1)^(1/2)-a],[2-a],x)