Internal problem ID [8827]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1247.
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 a x y^{\prime }+a \left (a -1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 27
dsolve((x^2-1)*diff(diff(y(x),x),x)+2*a*x*diff(y(x),x)+a*(a-1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (x +1\right )^{-a +1}+c_{2} \left (x -1\right )^{-a +1} \]
✓ Solution by Mathematica
Time used: 0.098 (sec). Leaf size: 90
DSolve[(-1 + a)*a*y[x] + 2*a*x*y'[x] + (-1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (1-x^2\right )^{\frac {1}{2}-\frac {1}{2} \sqrt {(a-1)^2}} \left (x^2-1\right )^{-a/2} \left (2 c_1 (1-x)^{\sqrt {(a-1)^2}}+\frac {c_2 (x+1)^{\sqrt {(a-1)^2}}}{\sqrt {(a-1)^2}}\right ) \\ \end{align*}