32.6.4 problem Exercise 12.4, page 103

Internal problem ID [5869]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous Methods
Problem number : Exercise 12.4, page 103
Date solved : Tuesday, March 04, 2025 at 11:49:41 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.053 (sec). Leaf size: 19

ode:=exp(y(x))*(diff(y(x),x)+1) = exp(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 1.259 (sec). Leaf size: 22

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

\[ y(x)\to -x+\log \left (\frac {e^{2 x}}{2}+c_1\right ) \]

✓ Sympy. Time used: 0.595 (sec). Leaf size: 14

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

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