29.15.13 problem 421

Internal problem ID [5019]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 15
Problem number : 421
Date solved : Sunday, March 30, 2025 at 06:30:12 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.042 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 4.813 (sec). Leaf size: 63

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

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

✓ Sympy. Time used: 0.929 (sec). Leaf size: 24

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

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