73.6.10 problem 7.4 (h)

Internal problem ID [15078]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 7. The exact form and general integrating fators. Additional exercises. page 141
Problem number : 7.4 (h)
Date solved : Monday, March 31, 2025 at 01:22:43 PM
CAS classification : [[_1st_order, _with_exponential_symmetries], _exact]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 13

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

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

✓ Mathematica. Time used: 4.451 (sec). Leaf size: 17

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

\[ y(x)\to c_1-W\left (e^{c_1} x\right ) \]

✓ Sympy. Time used: 0.658 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = - C_{1} - W\left (x e^{- C_{1}}\right ) \]