Internal problem ID [8811]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1231.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer]
Solve \begin {gather*} \boxed {\left (x^{2}-1\right ) y^{\prime \prime }-v \left (v +1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 57
dsolve((x^2-1)*diff(diff(y(x),x),x)-v*(v+1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (-x^{2}+1\right ) \hypergeom \left (\left [1+\frac {v}{2}, \frac {1}{2}-\frac {v}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right )+c_{2} \left (-x^{3}+x \right ) \hypergeom \left (\left [1-\frac {v}{2}, \frac {3}{2}+\frac {v}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.041 (sec). Leaf size: 56
DSolve[-(v*(1 + v)*y[x]) + (-1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \, _2F_1\left (\frac {1}{2} (-v-1),\frac {v}{2};\frac {1}{2};x^2\right )+i c_2 x \, _2F_1\left (-\frac {v}{2},\frac {v+1}{2};\frac {3}{2};x^2\right ) \\ \end{align*}