14.2.18 problem 19

Internal problem ID [2506]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 1. First order differential equations. Section 1.4 separable equations. Excercises page 24
Problem number : 19
Date solved : Monday, January 27, 2025 at 05:56:19 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 20

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

\[ y = -\frac {t}{\operatorname {LambertW}\left (\frac {c_1 t}{c_1 t -1}\right )} \]

✓ Solution by Mathematica

Time used: 1.390 (sec). Leaf size: 34

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

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