31.2.5 problem 5

Internal problem ID [5726]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 3
Problem number : 5
Date solved : Monday, January 27, 2025 at 01:11:39 PM
CAS classification : [[_homogeneous, `class A`], _exact, _dAlembert]

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

✓ Solution by Maple

Time used: 0.074 (sec). Leaf size: 20

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

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

✓ Solution by Mathematica

Time used: 1.308 (sec). Leaf size: 34

DSolve[(1+Exp[x/y[x]])+Exp[x/y[x]]*(1-x/y[x])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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