Internal problem ID [8821]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1241.
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 }-2 x y^{\prime }-\left (v +2\right ) \left (v -1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 29
dsolve((x^2-1)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)-(v+2)*(v-1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (x -1\right ) \left (x +1\right ) \LegendreP \left (v , 2, x\right )+c_{2} \left (x -1\right ) \left (x +1\right ) \LegendreQ \left (v , 2, x\right ) \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 26
DSolve[(1 - v)*(2 + v)*y[x] - 2*x*y'[x] + (-1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (x^2-1\right ) (c_1 P_v^2(x)+c_2 Q_v^2(x)) \\ \end{align*}