Internal problem ID [8977]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1398.
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 (3 x^{2}-1\right ) y^{\prime }}{\left (x^{2}-1\right ) x}+\frac {\left (x^{2}-1-\left (1+2 v \right )^{2}\right ) y}{\left (x^{2}-1\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 69
dsolve(diff(diff(y(x),x),x) = -1/(x^2-1)*(3*x^2-1)/x*diff(y(x),x)-(x^2-1-(2*v+1)^2)/(x^2-1)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (x^{2}-1\right )^{-\frac {1}{2}-v} \hypergeom \left (\left [-v , -v \right ], \left [-2 v \right ], -x^{2}+1\right )+c_{2} \left (x^{2}-1\right )^{v +\frac {1}{2}} \hypergeom \left (\left [v +1, v +1\right ], \left [2 v +2\right ], -x^{2}+1\right ) \]
✓ Solution by Mathematica
Time used: 0.101 (sec). Leaf size: 72
DSolve[y''[x] == -(((-1 - (1 + 2*v)^2 + x^2)*y[x])/(-1 + x^2)^2) - ((-1 + 3*x^2)*y'[x])/(x*(-1 + x^2)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \left (x^2-1\right )^{-v-\frac {1}{2}} \, _2F_1\left (-v,-v;-2 v;1-x^2\right )+c_2 \left (x^2-1\right )^{v+\frac {1}{2}} \, _2F_1\left (v+1,v+1;2 (v+1);1-x^2\right ) \\ \end{align*}