15.8.23 problem 24

Internal problem ID [3026]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 12, page 46
Problem number : 24
Date solved : Monday, January 27, 2025 at 07:09:19 AM
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: 0.050 (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)
 

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

✓ Solution by Mathematica

Time used: 1.344 (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*}