47.1.12 problem 12

Internal problem ID [7393]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.1 Separable equations problems. page 7
Problem number : 12
Date solved : Sunday, March 30, 2025 at 11:56:36 AM
CAS classification : [_quadrature]

\begin{align*} \left (1+z^{\prime }\right ) {\mathrm e}^{-z}&=1 \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 15

ode:=(1+diff(z(t),t))*exp(-z(t)) = 1; 
dsolve(ode,z(t), singsol=all);
 

\[ z = \ln \left (-\frac {1}{{\mathrm e}^{t} c_1 -1}\right ) \]

✓ Mathematica. Time used: 0.791 (sec). Leaf size: 28

ode=(1+D[z[t],t])*Exp[-z[t]]==1; 
ic={}; 
DSolve[{ode,ic},z[t],t,IncludeSingularSolutions->True]
 

\begin{align*} z(t)\to \log \left (\frac {1}{2} \left (1-\tanh \left (\frac {t+c_1}{2}\right )\right )\right ) \\ z(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.350 (sec). Leaf size: 14

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

\[ t + z{\left (t \right )} - \log {\left (e^{z{\left (t \right )}} - 1 \right )} = C_{1} \]