Internal problem ID [8957]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1378.
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 \left (x^{2}-1\right ) y^{\prime }}{x \left (x -1\right )^{2}}+\frac {\left (-2 x^{2}+2 x +2\right ) y}{x^{2} \left (x -1\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 50
dsolve(diff(diff(y(x),x),x) = -2/x*(x^2-1)/(x-1)^2*diff(y(x),x)-(-2*x^2+2*x+2)/x^2/(x-1)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} x^{2}}{x -1}+\frac {c_{2} x \left (\left (-x^{2}+x \right ) \ln \left (x -1\right )+\left (x^{2}-x \right ) \ln \relax (x )-x +\frac {1}{2}\right )}{\left (x -1\right )^{2}} \]
✓ Solution by Mathematica
Time used: 0.027 (sec). Leaf size: 41
DSolve[y''[x] == -(((2 + 2*x - 2*x^2)*y[x])/((-1 + x)^2*x^2)) - (2*(-1 + x^2)*y'[x])/((-1 + x)^2*x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x \left (c_2 \left (2 x+4 (x-1) x \tanh ^{-1}(1-2 x)-1\right )-c_1 (x-1) x\right )}{(x-1)^2} \\ \end{align*}