47.1.14 problem 14

Internal problem ID [7395]
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 : 14
Date solved : Monday, January 27, 2025 at 02:52:00 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.122 (sec). Leaf size: 19

dsolve([exp(x)-(1+exp(x))*y(x)*diff(y(x),x)=0,y(0) = 1],y(x), singsol=all)
 

\[ y = \sqrt {2 \ln \left ({\mathrm e}^{x}+1\right )-2 \ln \left (2\right )+1} \]

✓ Solution by Mathematica

Time used: 0.181 (sec). Leaf size: 23

DSolve[{Exp[x]-(1+Exp[x])*y[x]*D[y[x],x]==0,{y[0]==1}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \sqrt {2 \log \left (e^x+1\right )+1-\log (4)} \]