Internal problem ID [8832]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1252.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x \left (x +1\right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+y c=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 124
dsolve(x*(x+1)*diff(diff(y(x),x),x)+(a*x+b)*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [-\frac {1}{2}-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+\frac {a}{2}, -\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+\frac {a}{2}\right ], \left [a -b \right ], x +1\right )+c_{2} \left (x +1\right )^{-a +b +1} \hypergeom \left (\left [\frac {1}{2}-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}-\frac {a}{2}+b , \frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}-\frac {a}{2}+b \right ], \left [2-a +b \right ], x +1\right ) \]
✓ Solution by Mathematica
Time used: 0.076 (sec). Leaf size: 123
DSolve[c*y[x] + (b + a*x)*y'[x] + x*(1 + x)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 x^{1-b} \, _2F_1\left (\frac {1}{2} \left (a-2 b-\sqrt {(a-1)^2-4 c}+1\right ),\frac {1}{2} \left (a-2 b+\sqrt {(a-1)^2-4 c}+1\right );2-b;-x\right )+c_1 \, _2F_1\left (\frac {1}{2} \left (a-\sqrt {(a-1)^2-4 c}-1\right ),\frac {1}{2} \left (a+\sqrt {(a-1)^2-4 c}-1\right );b;-x\right ) \\ \end{align*}