Internal problem ID [8825]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1245.
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 \left (n -1\right ) x y^{\prime }-\left (v -n +1\right ) \left (v +n \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 35
dsolve((x^2-1)*diff(diff(y(x),x),x)-2*(n-1)*x*diff(y(x),x)-(v-n+1)*(v+n)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (x^{2}-1\right )^{\frac {n}{2}} \LegendreP \left (v , n , x\right )+c_{2} \left (x^{2}-1\right )^{\frac {n}{2}} \LegendreQ \left (v , n , x\right ) \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 32
DSolve[(-1 + n - v)*(n + v)*y[x] - 2*(-1 + n)*x*y'[x] + (-1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (x^2-1\right )^{n/2} (c_1 P_v^n(x)+c_2 Q_v^n(x)) \\ \end{align*}