68.6.28 problem 29

Internal problem ID [17338]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.4, page 57
Problem number : 29
Date solved : Thursday, October 02, 2025 at 02:05:54 PM
CAS classification : [[_homogeneous, `class A`], _exact, _dAlembert]

\begin{align*} -\frac {y^{2} {\mathrm e}^{\frac {y}{t}}}{t^{2}}+1+{\mathrm e}^{\frac {y}{t}} \left (1+\frac {y}{t}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 20

ode:=-1/t^2*y(t)^2*exp(y(t)/t)+1+exp(y(t)/t)*(1+y(t)/t)*diff(y(t),t) = 0; 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 22.461 (sec). Leaf size: 18

ode=(-t^(-2)*y[t]^2*Exp[y[t]/t]+1 )+Exp[y[t]/t]*(1+y[t]/t )*D[y[t],t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.859 (sec). Leaf size: 10

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((1 + y(t)/t)*exp(y(t)/t)*Derivative(y(t), t) + 1 - y(t)**2*exp(y(t)/t)/t**2,0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = t W\left (\frac {C_{1}}{t} - 1\right ) \]