49.22.3 problem 1(c)

Internal problem ID [7749]
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 : Monday, January 27, 2025 at 03:12:01 PM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 13

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

\[ y = \operatorname {LambertW}\left (-c_{1} -{\mathrm e}^{x}\right ) \]

✓ Solution by Mathematica

Time used: 60.160 (sec). Leaf size: 14

DSolve[Exp[x]+(Exp[y[x]]*(y[x]+1))*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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