85.33.85 problem 86

Internal problem ID [22708]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 65
Problem number : 86
Date solved : Thursday, October 02, 2025 at 09:11:18 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {{\mathrm e}^{x -y}}{y} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 14

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

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

✓ Mathematica. Time used: 60.075 (sec). Leaf size: 18

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

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

✓ Sympy. Time used: 0.173 (sec). Leaf size: 12

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

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