Internal problem ID [8879]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1300.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer]
Solve \begin {gather*} \boxed {\left (a^{2} x^{2}-1\right ) y^{\prime \prime }+2 a^{2} x y^{\prime }-2 a^{2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 32
dsolve((a^2*x^2-1)*diff(diff(y(x),x),x)+2*a^2*x*diff(y(x),x)-2*a^2*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = x c_{1}+c_{2} \left (\frac {a \ln \left (a x -1\right ) x}{2}-\frac {a \ln \left (a x +1\right ) x}{2}+1\right ) \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 23
DSolve[-2*a^2*y[x] + 2*a^2*x*y'[x] + (-1 + a^2*x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to a c_1 x+c_2 \left (a x \tanh ^{-1}(a x)-1\right ) \\ \end{align*}