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 : Sunday, March 30, 2025 at 11:37:34 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 24

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

\[ y = \operatorname {LambertW}\left ({\mathrm e}^{-1} \left (-\frac {{\mathrm e}^{-3 x}}{3}+c_1 -{\mathrm e}^{-x}\right )\right )+1 \]

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

ode=Exp[x]*y[x]*D[y[x],x]==Exp[-y[x]]+Exp[-2*x-y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.957 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*exp(x)*Derivative(y(x), x) - exp(-2*x - y(x)) - exp(-y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = W\left (C_{1} - \frac {e^{- 3 x - 1}}{3} - e^{- x - 1}\right ) + 1 \]