Internal problem ID [8838]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1258.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Jacobi]
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: 0.031 (sec). Leaf size: 110
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)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [-\frac {1}{2}-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+\frac {a}{2}, -\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+\frac {a}{2}\right ], \left [-b \right ], x\right )+c_{2} x^{b +1} \hypergeom \left (\left [\frac {1}{2}-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+\frac {a}{2}+b , \frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+\frac {a}{2}+b \right ], \left [b +2\right ], x\right ) \]
✓ Solution by Mathematica
Time used: 0.081 (sec). Leaf size: 121
DSolve[c*y[x] + (b + a*x)*y'[x] + (-1 + x)*x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 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 )-(-1)^b c_2 x^{b+1} \, _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 );b+2;x\right ) \\ \end{align*}