38.4.46 problem 28

Internal problem ID [8345]
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 2.1 Solution curves without a solution. Exercises 2.1 at page 44
Problem number : 28
Date solved : Tuesday, September 30, 2025 at 05:29:27 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=\left (y \,{\mathrm e}^{y}-9 y\right ) {\mathrm e}^{-y} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 25

ode:=diff(y(x),x) = (y(x)*exp(y(x))-9*y(x))/exp(y(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ x +\int _{}^{y}\frac {1}{\textit {\_a} \left (9 \,{\mathrm e}^{-\textit {\_a}}-1\right )}d \textit {\_a} +c_1 = 0 \]

✓ Mathematica. Time used: 0.373 (sec). Leaf size: 47

ode=D[y[x],x]==(y[x]*Exp[y[x]]-9*y[x])/Exp[y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{K[1]}}{\left (-9+e^{K[1]}\right ) K[1]}dK[1]\&\right ][x+c_1]\\ y(x)&\to 0\\ y(x)&\to \log (9) \end{align*}

✓ Sympy. Time used: 0.869 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-y(x)*exp(y(x)) + 9*y(x))*exp(-y(x)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ - \int \limits ^{y{\left (x \right )}} \frac {e^{y}}{y \left (e^{y} - 9\right )}\, dy = C_{1} - x \]