54.2.191 problem 768

Internal problem ID [12065]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 768
Date solved : Wednesday, October 01, 2025 at 12:09:00 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{\prime }&=\frac {y \left (y+1\right )}{x \left (-y-1+x y\right )} \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 26

ode:=diff(y(x),x) = y(x)*(1+y(x))/x/(-y(x)-1+x*y(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {1}{x \operatorname {LambertW}\left (\frac {{\mathrm e}^{-\frac {1}{x}}}{x c_1}\right )+1} \]

✓ Mathematica. Time used: 0.173 (sec). Leaf size: 72

ode=D[y[x],x] == (y[x]*(1 + y[x]))/(x*(-1 - y[x] + x*y[x])); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{-\frac {2 x y(x)+y(x)+1}{\sqrt [3]{2} ((x-1) y(x)-1)}}\frac {1}{K[1]^3-\frac {3 K[1]}{2^{2/3}}+1}dK[1]+\frac {1}{9} 2^{2/3} \left (\frac {1}{x}+\log (x)-1\right )=c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.634 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (y(x) + 1)*y(x)/(x*(x*y(x) - y(x) - 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} + \log {\left (y{\left (x \right )} + 1 \right )} - \log {\left (y{\left (x \right )} \right )} - \frac {1}{x y{\left (x \right )}} = 0 \]