44.5.8 problem 8

Internal problem ID [7070]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.2 Separable equations. Exercises 2.2 at page 53
Problem number : 8
Date solved : Tuesday, January 28, 2025 at 03:10:40 PM
CAS classification : [_separable]

\begin{align*} {\mathrm e}^{x} y y^{\prime }&={\mathrm e}^{-y}+{\mathrm e}^{-2 x -y} \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 30

dsolve(exp(x)*y(x)*diff(y(x),x)=exp(-y(x))+exp(-2*x-y(x)),y(x), singsol=all)
 

\[ y = \operatorname {LambertW}\left (\frac {\left (3 \,{\mathrm e}^{3 x} c_{1} -3 \,{\mathrm e}^{2 x}-1\right ) {\mathrm e}^{-3 x -1}}{3}\right )+1 \]

✓ Solution by Mathematica

Time used: 60.177 (sec). Leaf size: 37

DSolve[Exp[x]*y[x]*D[y[x],x]==Exp[-y[x]]+Exp[-2*x-y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to 1+W\left (\frac {1}{3} e^{-3 x-1} \left (-3 e^{2 x}+3 c_1 e^{3 x}-1\right )\right ) \]