Internal problem ID [8819]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1239.
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 }-l 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*x*diff(y(x),x)-l*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \LegendreP \left (\frac {\sqrt {1+4 l}}{2}-\frac {1}{2}, x\right )+c_{2} \LegendreQ \left (\frac {\sqrt {1+4 l}}{2}-\frac {1}{2}, x\right ) \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 46
DSolve[-(l*y[x]) + 2*x*y'[x] + (-1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 P_{\frac {1}{2} \left (\sqrt {4 l+1}-1\right )}(x)+c_2 Q_{\frac {1}{2} \left (\sqrt {4 l+1}-1\right )}(x) \\ \end{align*}