26.5.20 problem 25

Internal problem ID [4294]
Book : Differential equations with applications and historial notes, George F. Simmons. Second edition. 1971
Section : Chapter 2, End of chapter, page 61
Problem number : 25
Date solved : Sunday, March 30, 2025 at 02:52:37 AM
CAS classification : [`y=_G(x,y')`]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.314 (sec). Leaf size: 26

ode=Exp[x]*(1+x)==(x*Exp[x]-y[x]*Exp[y[x]])*D[y[x],x]; 
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: 1.484 (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)
 

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