Internal problem ID [8456]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 876.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class C]]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {y^{2} \left (x^{2} y-2 x -2 x y+y\right )}{2 \left (-2+x y-2 y\right ) x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.001 (sec). Leaf size: 41
dsolve(diff(y(x),x) = -1/2*y(x)^2*(x^2*y(x)-2*x-2*x*y(x)+y(x))/(-2+x*y(x)-2*y(x))/x,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {4}{\sqrt {c_{1}-8 \ln \relax (x )}+2 x -4} \\ y \relax (x ) = -\frac {4}{\sqrt {c_{1}-8 \ln \relax (x )}-2 x +4} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.27 (sec). Leaf size: 90
DSolve[y'[x] == -1/2*(y[x]^2*(-2*x + y[x] - 2*x*y[x] + x^2*y[x]))/(x*(-2 - 2*y[x] + x*y[x])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2}{x+\sqrt {2} \sqrt {-\frac {1}{x}} \sqrt {x (\log (x)-2 (1+c_1))}-2} \\ y(x)\to -\frac {2}{-x+\sqrt {2} \sqrt {-\frac {1}{x}} \sqrt {x (\log (x)-2 (1+c_1))}+2} \\ y(x)\to 0 \\ \end{align*}