54.2.349 problem 928

Internal problem ID [12223]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 928
Date solved : Sunday, October 12, 2025 at 02:27:26 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 20

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

\[ y = -\ln \left (\frac {-\ln \left (x +1\right )+c_1}{x}\right ) x \]

✓ Mathematica. Time used: 1.176 (sec). Leaf size: 22

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

\begin{align*} y(x)&\to -x \log \left (\frac {-\log (x+1)+c_1}{x}\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (x**2 + x*y(x) + x*exp(y(x)/x) + x + y(x))/(x*(x + 1)) cannot be solved by the factorable group method