15.33 problem 441

Internal problem ID [3187]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 15
Problem number: 441.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [[_homogeneous, class A], _dAlembert]

Solve \begin {gather*} \boxed {\left (x -y\right ) y^{\prime }-\left ({\mathrm e}^{-\frac {x}{y}}+1\right ) y=0} \end {gather*}

Solution by Maple

Time used: 0.019 (sec). Leaf size: 20

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

\[ y \relax (x ) = -\frac {x}{\LambertW \left (\frac {x c_{1}}{c_{1} x -1}\right )} \]

Solution by Mathematica

Time used: 1.447 (sec). Leaf size: 34

DSolve[(x-y[x])y'[x]==(Exp[-x/y[x]]+1)y[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {x}{\text {ProductLog}\left (\frac {x}{x-e^{c_1}}\right )} \\ y(x)\to -e^{\text {ProductLog}(1)} x \\ \end{align*}