Internal problem ID [8940]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1361.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\frac {2 x y^{\prime }}{x^{2}-1}+\frac {\left (a \left (a +1\right )-a \,x^{2} \left (3+a \right )\right ) y}{x^{2} \left (x^{2}-1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 33
dsolve(diff(diff(y(x),x),x) = 2*x/(x^2-1)*diff(y(x),x)-(a*(a+1)-a*x^2*(a+3))/x^2/(x^2-1)*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{-a}+c_{2} \left (2 a \,x^{2}+x^{2}-2 a -3\right ) x^{a +1} \]
✓ Solution by Mathematica
Time used: 0.241 (sec). Leaf size: 36
DSolve[y''[x] == -(((a*(1 + a) - a*(3 + a)*x^2)*y[x])/(x^2*(-1 + x^2))) + (2*x*y'[x])/(-1 + x^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 x^{-a}-c_2 x^{a+1} \left (2 a \left (x^2-1\right )+x^2-3\right ) \\ \end{align*}