Internal problem ID [8938]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1359.
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} \left (x^{2}-1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.03 (sec). Leaf size: 57
dsolve(diff(diff(y(x),x),x) = -2*x/(x^2-1)*diff(y(x),x)-v*(v+1)/x^2/(x^2-1)*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [-\frac {v}{2}, \frac {1}{2}-\frac {v}{2}\right ], \left [\frac {1}{2}-v \right ], x^{2}\right ) x^{-v}+c_{2} \hypergeom \left (\left [1+\frac {v}{2}, \frac {1}{2}+\frac {v}{2}\right ], \left [\frac {3}{2}+v \right ], x^{2}\right ) x^{v +1} \]
✓ Solution by Mathematica
Time used: 0.059 (sec). Leaf size: 84
DSolve[y''[x] == -((v*(1 + v)*y[x])/(x^2*(-1 + 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-v}{2},-\frac {v}{2};\frac {1}{2}-v;x^2\right )+c_2 i^{v+1} x^{v+1} \, _2F_1\left (\frac {v+1}{2},\frac {v+2}{2};v+\frac {3}{2};x^2\right ) \\ \end{align*}