23.1.745 problem 748

Internal problem ID [5352]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 748
Date solved : Tuesday, September 30, 2025 at 12:35:18 PM
CAS classification : [_exact]

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

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

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

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

✓ Mathematica. Time used: 1.884 (sec). Leaf size: 33

ode=(Exp[x]+x*Exp[y[x]])*D[y[x],x]+y[x]*Exp[x]+Exp[y[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