83.8.3 problem 3

Internal problem ID [21956]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter III. First order differential equations of the first degree. Ex. IX at page 55
Problem number : 3
Date solved : Thursday, October 02, 2025 at 08:19:45 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.081 (sec). Leaf size: 23

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

\[ y = \frac {-\operatorname {LambertW}\left ({\mathrm e}^{\frac {1}{x c_1}}\right ) x c_1 +1}{c_1} \]

✓ Mathematica. Time used: 1.76 (sec). Leaf size: 33

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

\begin{align*} y(x)&\to e^{c_1}-x W\left (e^{\frac {e^{c_1}}{x}}\right )\\ y(x)&\to W(1) (-x) \end{align*}

✓ Sympy. Time used: 1.041 (sec). Leaf size: 12

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

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