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 : Monday, January 27, 2025 at 10:03:53 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 25

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

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

✓ Solution by Mathematica

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

DSolve[y[x] D[y[x],x]+x Exp[-x](1+y[x])==0,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*}