68.7.32 problem 32

Internal problem ID [17396]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.5, page 64
Problem number : 32
Date solved : Thursday, October 02, 2025 at 02:16:42 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} t \left (\ln \left (t \right )-\ln \left (y\right )\right ) y^{\prime }&=y \end{align*}

✓ Maple. Time used: 0.085 (sec). Leaf size: 14

ode:=t*(ln(t)-ln(y(t)))*diff(y(t),t) = y(t); 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 2.289 (sec). Leaf size: 37

ode=t*(Log[t]-Log[y[t]] )*D[y[t],t]==y[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -e^{c_1} W\left (-e^{-1-c_1} t\right )\\ y(t)&\to 0\\ y(t)&\to \frac {t}{e} \end{align*}

✓ Sympy. Time used: 0.736 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*(log(t) - log(y(t)))*Derivative(y(t), t) - y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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