54.2.107 problem 683

Internal problem ID [11981]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 683
Date solved : Tuesday, September 30, 2025 at 11:51:27 PM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }&=\frac {y \left (-1+\ln \left (x \left (x +1\right )\right ) y x^{4}-\ln \left (x \left (x +1\right )\right ) x^{3}\right )}{x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 45

ode:=diff(y(x),x) = y(x)*(-1+ln(x*(1+x))*y(x)*x^4-ln(x*(1+x))*x^3)/x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {1}{x \left (\left (x \left (x +1\right )\right )^{\frac {x^{3}}{3}} c_1 \left (x +1\right )^{{1}/{3}} {\mathrm e}^{-\frac {2}{9} x^{3}+\frac {1}{6} x^{2}-\frac {1}{3} x}+1\right )} \]

✓ Mathematica. Time used: 1.823 (sec). Leaf size: 232

ode=D[y[x],x] == (y[x]*(-1 - x^3*Log[x*(1 + x)] + x^4*Log[x*(1 + x)]*y[x]))/x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^{\frac {1}{18} x \left (4 x^2-3 x+6\right )} (x+1)^{2/3} (x (x+1))^{-\frac {x^3}{3}-1}}{-\int _1^x\frac {e^{\frac {1}{18} K[1] \left (4 K[1]^2-3 K[1]+6\right )} K[1]^2 (K[1] (K[1]+1))^{-\frac {1}{3} K[1]^3} \log (K[1] (K[1]+1))}{\sqrt [3]{K[1]+1}}dK[1]+c_1}\\ y(x)&\to 0\\ y(x)&\to -\frac {e^{\frac {1}{18} x \left (4 x^2-3 x+6\right )} (x+1)^{2/3} (x (x+1))^{-\frac {x^3}{3}-1}}{\int _1^x\frac {e^{\frac {1}{18} K[1] \left (4 K[1]^2-3 K[1]+6\right )} K[1]^2 (K[1] (K[1]+1))^{-\frac {1}{3} K[1]^3} \log (K[1] (K[1]+1))}{\sqrt [3]{K[1]+1}}dK[1]} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**4*y(x)*log(x*(x + 1)) - x**3*log(x*(x + 1)) - 1)*y(x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out