9.1.26 problem 26

Internal problem ID [2866]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 5, page 21
Problem number : 26
Date solved : Tuesday, September 30, 2025 at 05:55:32 AM
CAS classification : [_quadrature]

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

With initial conditions

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

✓ Maple. Time used: 0.315 (sec). Leaf size: 22

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

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 18

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

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

✓ Sympy. Time used: 0.150 (sec). Leaf size: 14

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

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