60.2.299 problem 876

Internal problem ID [10886]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 876
Date solved : Monday, January 27, 2025 at 10:21:10 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=-\frac {y^{2} \left (x^{2} y-2 x -2 y x +y\right )}{2 \left (-2+y x -2 y\right ) x} \end{align*}

✓ Solution by Maple

Time used: 0.002 (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 &= \frac {4}{\sqrt {c_{1} -8 \ln \left (x \right )}+2 x -4} \\ y &= -\frac {4}{\sqrt {c_{1} -8 \ln \left (x \right )}-2 x +4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.884 (sec). Leaf size: 94

DSolve[D[y[x],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+2 c_1)}-2} \\ y(x)\to -\frac {2}{-x+\sqrt {2} \sqrt {-\frac {1}{x}} \sqrt {-x (-\log (x)+2+2 c_1)}+2} \\ y(x)\to 0 \\ \end{align*}