54.2.297 problem 876

Internal problem ID [12171]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 876
Date solved : Wednesday, October 01, 2025 at 01:06:05 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class C`]]

\begin{align*} 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} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 45

ode:=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; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ 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*}

✓ Mathematica. Time used: 0.631 (sec). Leaf size: 94

ode=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])); 
ic={}; 
DSolve[{ode,ic},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*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) + (x**2*y(x) - 2*x*y(x) - 2*x + y(x))*y(x)**2/(2*x*(x*y(x) - 2*y(x) - 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out