87.5.7 problem 7

Internal problem ID [23300]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 1. Elementary methods. First order differential equations. Exercise at page 47
Problem number : 7
Date solved : Thursday, October 02, 2025 at 09:29:11 PM
CAS classification : [_exact, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y \,{\mathrm e}^{y x}+\left (x \,{\mathrm e}^{y x}+1\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 22

ode:=y(x)*exp(x*y(x))+(x*exp(x*y(x))+1)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 2.648 (sec). Leaf size: 27

ode=y[x]*Exp[x*y[x]]+(x*Exp[x*y[x]]+1)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1-\frac {W\left (x e^{c_1 x}\right )}{x}\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) + y(x)*exp(x*y(x))/(x*exp(x*y(x)) + 1) canno