54.2.145 problem 722

Internal problem ID [12019]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 722
Date solved : Tuesday, September 30, 2025 at 11:56:36 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=-\frac {y^{3}}{\left (-1+2 y \ln \left (x \right )-y\right ) x} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 45

ode:=diff(y(x),x) = -y(x)^3/(-1+2*y(x)*ln(x)-y(x))/x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {1}{{\mathrm e}^{-\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} \ln \left (\frac {{\mathrm e}^{\textit {\_Z}}+2}{x^{4}}\right )+{\mathrm e}^{\textit {\_Z}} \ln \left (2\right )+3 c_1 \,{\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2\right )}+2 \ln \left (x \right )-1} \]

✓ Mathematica. Time used: 0.268 (sec). Leaf size: 122

ode=D[y[x],x] == -(y[x]^3/(x*(-1 - y[x] + 2*Log[x]*y[x]))); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{-\frac {(1-2 \log (x))^2 \left (-\frac {1}{(2 \log (x)-1)^3}\right )^{2/3} ((5-4 \log (x)) y(x)+2)}{2 \sqrt [3]{2} ((2 \log (x)-1) y(x)-1)}}\frac {2}{2 K[1]^3+3 \sqrt [3]{-2} K[1]+2}dK[1]=\frac {4}{9} 2^{2/3} \log (x) \left (-\frac {1}{(2 \log (x)-1)^3}\right )^{2/3} (1-2 \log (x))^2+c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.197 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) + y(x)**3/(x*(2*y(x)*log(x) - y(x) - 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} - \frac {\left (3 y{\left (x \right )} + 2\right ) e^{- \frac {2}{y{\left (x \right )}}}}{4 y{\left (x \right )}} + e^{- \frac {2}{y{\left (x \right )}}} \log {\left (x \right )} = 0 \]