38.5.13 problem 13

Internal problem ID [8361]
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 : 13
Date solved : Tuesday, September 30, 2025 at 05:30:02 PM
CAS classification : [_separable]

\begin{align*} \left ({\mathrm e}^{y}+1\right )^{2} {\mathrm e}^{-y}+\left ({\mathrm e}^{x}+1\right )^{3} {\mathrm e}^{-x} y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 31

ode:=(exp(y(x))+1)^2*exp(-y(x))+(exp(x)+1)^3*exp(-x)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -2 \,\operatorname {arctanh}\left (\frac {2 \,{\mathrm e}^{2 x} c_1 +4 \,{\mathrm e}^{x} c_1 +2 c_1 -1}{\left ({\mathrm e}^{x}+1\right )^{2}}\right ) \]

✓ Mathematica. Time used: 4.146 (sec). Leaf size: 35

ode=(Exp[y[x]]+1)^2*Exp[-y[x]]+(Exp[x]+1)^3*Exp[-x]*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \log \left (-1-\frac {1}{\frac {1}{2 \left (e^x+1\right )^2}+c_1}\right )\\ y(x)&\to i \pi \end{align*}

✓ Sympy. Time used: 0.391 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((exp(x) + 1)**3*exp(-x)*Derivative(y(x), x) + (exp(y(x)) + 1)**2*exp(-y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \log {\left (\frac {- 2 C_{1} e^{2 x} - 4 C_{1} e^{x} - 2 C_{1} - 2 e^{2 x} - 4 e^{x} - 3}{2 C_{1} e^{2 x} + 4 C_{1} e^{x} + 2 C_{1} + 1} \right )} \]