Internal problem ID [8836]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1256.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Jacobi]
Solve \begin {gather*} \boxed {x \left (x -1\right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }-v \left (v +1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 51
dsolve(x*(x-1)*diff(diff(y(x),x),x)+(2*x-1)*diff(y(x),x)-v*(v+1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [-v , -v \right ], \left [-2 v \right ], \frac {1}{x}\right ) x^{v}+c_{2} \hypergeom \left (\left [v +1, v +1\right ], \left [2 v +2\right ], \frac {1}{x}\right ) x^{-v -1} \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 26
DSolve[-(v*(1 + v)*y[x]) + (-1 + 2*x)*y'[x] + (-1 + x)*x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 P_v(2 x-1)+c_2 Q_v(2 x-1) \\ \end{align*}