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 : Monday, January 27, 2025 at 02:51:55 PM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 15

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

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

✓ Solution by Mathematica

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

DSolve[(1+D[z[t],t])*Exp[-z[t]]==1,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*}