54.1.334 problem 341

Internal problem ID [11648]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 341
Date solved : Tuesday, September 30, 2025 at 09:58:28 PM
CAS classification : [_exact]

\begin{align*} \left (x \,{\mathrm e}^{y}+{\mathrm e}^{x}\right ) y^{\prime }+{\mathrm e}^{y}+y \,{\mathrm e}^{x}&=0 \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 29

ode:=(x*exp(y(x))+exp(x))*diff(y(x),x)+exp(y(x))+y(x)*exp(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 1.963 (sec). Leaf size: 33

ode=E^y[x] + E^x*y[x] + (E^x + E^y[x]*x)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 e^{-x}-W\left (x e^{-x+c_1 e^{-x}}\right ) \end{align*}

✗ Sympy

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

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