Internal problem ID [8939]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1360.
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 {2 x y^{\prime }}{x^{2}-1}-\frac {v \left (v +1\right ) y}{x^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 47
dsolve(diff(diff(y(x),x),x) = -2*x/(x^2-1)*diff(y(x),x)+v*(v+1)/x^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [\frac {1}{2}, -v \right ], \left [\frac {1}{2}-v \right ], x^{2}\right ) x^{-v}+c_{2} \hypergeom \left (\left [\frac {1}{2}, v +1\right ], \left [\frac {3}{2}+v \right ], x^{2}\right ) x^{v +1} \]
✓ Solution by Mathematica
Time used: 0.048 (sec). Leaf size: 68
DSolve[y''[x] == (v*(1 + v)*y[x])/x^2 - (2*x*y'[x])/(-1 + x^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 i^{-v} x^{-v} \, _2F_1\left (\frac {1}{2},-v;\frac {1}{2}-v;x^2\right )+c_2 i^{v+1} x^{v+1} \, _2F_1\left (\frac {1}{2},v+1;v+\frac {3}{2};x^2\right ) \\ \end{align*}