Internal problem ID [9688]
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]]
\[ \boxed {y^{\prime \prime }+\frac {2 x y^{\prime }}{x^{2}-1}-\frac {v \left (v +1\right ) y}{x^{2}}=0} \]
✓ Solution by Maple
Time used: 0.282 (sec). Leaf size: 109
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)
\[ y \left (x \right ) = \frac {c_{2} x^{v +1} \left (x^{2}\right )^{-\frac {v}{2}-\frac {1}{4}} \Gamma \left (v +\frac {1}{2}\right )^{2} \left (v +\frac {1}{2}\right ) \operatorname {LegendreP}\left (-\frac {1}{2}, -v -\frac {1}{2}, \frac {-x^{2}-1}{x^{2}-1}\right )+\sec \left (\pi v \right ) \pi \operatorname {LegendreP}\left (-\frac {1}{2}, v +\frac {1}{2}, \frac {-x^{2}-1}{x^{2}-1}\right ) \left (x^{2}\right )^{\frac {1}{4}+\frac {v}{2}} x^{-v} c_{1}}{\sqrt {-x^{2}+1}\, \Gamma \left (v +\frac {1}{2}\right )} \]
✓ Solution by Mathematica
Time used: 0.119 (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]
\[ y(x)\to c_1 i^{-v} x^{-v} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-v,\frac {1}{2}-v,x^2\right )+c_2 i^{v+1} x^{v+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},v+1,v+\frac {3}{2},x^2\right ) \]