26.4.18 problem Recognizable Exact Differential equations. Integrating factors. Exercise 10.10, page 90

Internal problem ID [6964]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 10
Problem number : Recognizable Exact Differential equations. Integrating factors. Exercise 10.10, page 90
Date solved : Tuesday, September 30, 2025 at 04:07:21 PM
CAS classification : [`y=_G(x,y')`]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 20

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

\[ x \,{\mathrm e}^{x -y}+\frac {y^{2}}{2}+c_1 = 0 \]

✓ Mathematica. Time used: 0.188 (sec). Leaf size: 26

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

\[ \text {Solve}\left [-\frac {1}{2} y(x)^2-x e^{x-y(x)}=c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.924 (sec). Leaf size: 19

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

\[ C_{1} + x e^{x} e^{- y{\left (x \right )}} + \frac {y^{2}{\left (x \right )}}{2} = 0 \]