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 = 0.153564 (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 = 0.069 (sec), leaf count = 124
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\mbox {$_2$F$_1$}(-{\frac {1}{2}}+{\frac {1}{2}\sqrt {{b}^{2}-2\,b-4\,c+1}}+{\frac {b}{2}},-{\frac {1}{2}}-{\frac {1}{2}\sqrt {{b}^{2}-2\,b-4\,c+1}}+{\frac {b}{2}};\,-a+b;\,1+x)}+{\it \_C2}\, \left ( 1+x \right ) ^{a-b+1}{\mbox {$_2$F$_1$}({\frac {1}{2}}-{\frac {1}{2}\sqrt {{b}^{2}-2\,b-4\,c+1}}-{\frac {b}{2}}+a,{\frac {1}{2}}+{\frac {1}{2}\sqrt {{b}^{2}-2\,b-4\,c+1}}-{\frac {b}{2}}+a;\,2+a-b;\,1+x)} \right \} \] 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*(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^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],[-a+b],1+x)+_C2*(1+x)^(a-b+1)*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],1+x)