Internal problem ID [9349]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1770.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {x^{2} \left (-1+y\right ) y^{\prime \prime }-2 x^{2} \left (y^{\prime }\right )^{2}-2 x \left (-1+y\right ) y^{\prime }-2 y \left (-1+y\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.051 (sec). Leaf size: 30
dsolve(x^2*(-1+y(x))*diff(diff(y(x),x),x)-2*x^2*diff(y(x),x)^2-2*x*(-1+y(x))*diff(y(x),x)-2*y(x)*(-1+y(x))^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 1 \\ y \relax (x ) = \frac {x \left (x c_{1}-c_{2}\right )}{c_{1} x^{2}-x c_{2}-1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.51 (sec). Leaf size: 22
DSolve[-2*(-1 + y[x])^2*y[x] - 2*x*(-1 + y[x])*y'[x] - 2*x^2*y'[x]^2 + x^2*(-1 + y[x])*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 1+\frac {1}{c_2 x^2-c_1 x-1} \\ \end{align*}