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 : Sunday, March 30, 2025 at 11:56:41 AM
CAS classification : [_separable]

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

With initial conditions

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

✓ Maple. Time used: 0.161 (sec). Leaf size: 19

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

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

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

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

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

✓ Sympy. Time used: 0.629 (sec). Leaf size: 19

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

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