29.26.8 problem 744

Internal problem ID [5329]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 26
Problem number : 744
Date solved : Monday, January 27, 2025 at 11:14:52 AM
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*}

✓ Solution by Maple

Time used: 0.054 (sec). Leaf size: 29

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

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

✓ Solution by Mathematica

Time used: 2.126 (sec). Leaf size: 33

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

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