Internal problem ID [8904]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1325.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (\left (a +b +1\right ) x +\alpha +\beta -1\right ) y^{\prime }}{x \left (x -1\right )}+\frac {\left (a b x -\alpha \beta \right ) y}{x^{2} \left (x -1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 103
dsolve(diff(diff(y(x),x),x) = -((a+b+1)*x+alpha+beta-1)/x/(x-1)*diff(y(x),x)-(a*b*x-alpha*beta)/x^2/(x-1)*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (x -1\right )^{1-a -\alpha -b -\beta } x^{\alpha } \hypergeom \left (\left [1-b -\beta , 1-a -\beta \right ], \left [1+\alpha -\beta \right ], x\right )+c_{2} \left (x -1\right )^{1-a -\alpha -b -\beta } x^{\beta } \hypergeom \left (\left [1-\alpha -b , 1-a -\alpha \right ], \left [1-\alpha +\beta \right ], x\right ) \]
✓ Solution by Mathematica
Time used: 0.121 (sec). Leaf size: 52
DSolve[y''[x] == -(((-(\[Alpha]*\[Beta]) + a*b*x)*y[x])/((-1 + x)*x^2)) - ((-1 + \[Alpha] + \[Beta] + (1 + a + b)*x)*y'[x])/((-1 + x)*x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to (-1)^{\beta } c_2 x^{\beta } \, _2F_1(a+\beta ,b+\beta ;-\alpha +\beta +1;x)+(-1)^{\alpha } c_1 x^{\alpha } \, _2F_1(a+\alpha ,b+\alpha ;\alpha -\beta +1;x) \\ \end{align*}