Internal problem ID [8913]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1334.
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 (x \left (a +1\right )-1\right ) y^{\prime }}{x \left (x -1\right )}+\frac {\left (\left (a^{2}-b^{2}\right ) x +c^{2}\right ) y}{4 x^{2} \left (x -1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 97
dsolve(diff(diff(y(x),x),x) = -((a+1)*x-1)/x/(x-1)*diff(y(x),x)-1/4*((a^2-b^2)*x+c^2)/x^2/(x-1)*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (x -1\right )^{-a +1} x^{\frac {c}{2}} \hypergeom \left (\left [-\frac {a}{2}-\frac {b}{2}+\frac {c}{2}+1, -\frac {a}{2}+\frac {b}{2}+\frac {c}{2}+1\right ], \left [c +1\right ], x\right )+c_{2} \left (x -1\right )^{-a +1} x^{-\frac {c}{2}} \hypergeom \left (\left [-\frac {a}{2}-\frac {b}{2}-\frac {c}{2}+1, -\frac {a}{2}+\frac {b}{2}-\frac {c}{2}+1\right ], \left [1-c \right ], x\right ) \]
✓ Solution by Mathematica
Time used: 0.104 (sec). Leaf size: 89
DSolve[y''[x] == -1/4*((c^2 + (a^2 - b^2)*x)*y[x])/((-1 + x)*x^2) - ((-1 + (1 + a)*x)*y'[x])/((-1 + x)*x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to i^{-c} x^{-c/2} \left (i^{2 c} c_2 x^c \, _2F_1\left (\frac {1}{2} (a-b+c),\frac {1}{2} (a+b+c);c+1;x\right )+c_1 \, _2F_1\left (\frac {1}{2} (a-b-c),\frac {1}{2} (a+b-c);1-c;x\right )\right ) \\ \end{align*}