81.1.34 problem 2-32

Internal problem ID [21479]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 2. Separable differential equations
Problem number : 2-32
Date solved : Thursday, October 02, 2025 at 07:40:01 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 28

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

\[ y = \ln \left (2\right )+\ln \left (-\frac {1}{x^{2} {\mathrm e}^{-c_1}-1}\right )-x -c_1 \]

✓ Mathematica. Time used: 0.229 (sec). Leaf size: 24

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

\begin{align*} y(x)&\to -x-\log \left (-\frac {x^2}{2}-c_1\right ) \end{align*}

✓ Sympy. Time used: 0.351 (sec). Leaf size: 15

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

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