43.22.3 problem 1(c)

Internal problem ID [9035]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 5. Existence and uniqueness of solutions to first order equations. Page 198
Problem number : 1(c)
Date solved : Tuesday, September 30, 2025 at 06:02:18 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 13

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

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

✓ Mathematica. Time used: 60.104 (sec). Leaf size: 14

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

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

✓ Sympy. Time used: 0.155 (sec). Leaf size: 8

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

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