Internal problem ID [8937]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1358.
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 {\left (x^{2}-2\right ) y^{\prime }}{x \left (x^{2}-1\right )}+\frac {\left (x^{2}-2\right ) y}{x^{2} \left (x^{2}-1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 21
dsolve(diff(diff(y(x),x),x) = 1/x*(x^2-2)/(x^2-1)*diff(y(x),x)-(x^2-2)/x^2/(x^2-1)*y(x),y(x), singsol=all)
\[ y \relax (x ) = x c_{1}+c_{2} x \ln \left (x +\sqrt {x^{2}-1}\right ) \]
✓ Solution by Mathematica
Time used: 0.031 (sec). Leaf size: 45
DSolve[y''[x] == -(((-2 + x^2)*y[x])/(x^2*(-1 + x^2))) + ((-2 + x^2)*y'[x])/(x*(-1 + x^2)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x \sqrt [4]{x^2-1} \left (c_2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-1}}\right )+c_1\right )}{\sqrt [4]{1-x^2}} \\ \end{align*}