ODE
\[ (a+b x) y'(x)+c y(x)+x (x+1) y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.28507 (sec), leaf count = 131
\[\left \{\left \{y(x)\to c_2 x^{1-a} \, _2F_1\left (\frac {1}{2} \left (-2 a+b-\sqrt {b^2-2 b-4 c+1}+1\right ),\frac {1}{2} \left (-2 a+b+\sqrt {b^2-2 b-4 c+1}+1\right );2-a;-x\right )+c_1 \, _2F_1\left (\frac {1}{2} \left (b-\sqrt {b^2-2 b-4 c+1}-1\right ),\frac {1}{2} \left (b+\sqrt {b^2-2 b-4 c+1}-1\right );a;-x\right )\right \}\right \}\]
Maple ✓
cpu = 0.374 (sec), leaf count = 124
\[\left [y \left (x \right ) = \textit {\_C1} \hypergeom \left (\left [-\frac {1}{2}+\frac {\sqrt {b^{2}-2 b -4 c +1}}{2}+\frac {b}{2}, -\frac {1}{2}-\frac {\sqrt {b^{2}-2 b -4 c +1}}{2}+\frac {b}{2}\right ], \left [b -a \right ], x +1\right )+\textit {\_C2} \left (x +1\right )^{1+a -b} \hypergeom \left (\left [\frac {1}{2}-\frac {\sqrt {b^{2}-2 b -4 c +1}}{2}-\frac {b}{2}+a , \frac {1}{2}+\frac {\sqrt {b^{2}-2 b -4 c +1}}{2}-\frac {b}{2}+a \right ], \left [2+a -b \right ], x +1\right )\right ]\] Mathematica raw input
DSolve[c*y[x] + (a + b*x)*y'[x] + x*(1 + 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] + 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*(x+1)*diff(diff(y(x),x),x)+(b*x+a)*diff(y(x),x)+c*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*hypergeom([-1/2+1/2*(b^2-2*b-4*c+1)^(1/2)+1/2*b, -1/2-1/2*(b^2-2*b-4*c+1)^(1/2)+1/2*b],[b-a],x+1)+_C2*(x+1)^(1+a-b)*hypergeom([1/2-1/2*(b^2-2*b-4*c+1)^(1/2)-1/2*b+a, 1/2+1/2*(b^2-2*b-4*c+1)^(1/2)-1/2*b+a],[2+a-b],x+1)]