68.7.13 problem 13

Internal problem ID [17377]
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 : 13
Date solved : Thursday, October 02, 2025 at 02:14:57 PM
CAS classification : [_separable]

\begin{align*} 2 \ln \left (t \right )-\ln \left (4 y^{2}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 59

ode:=2*ln(t)-ln(4*y(t)^2)*diff(y(t),t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\begin{align*} y &= \frac {t \ln \left (t \right )+c_1 -t}{\operatorname {LambertW}\left (-2 \left (t \ln \left (t \right )+c_1 -t \right ) {\mathrm e}^{-1}\right )} \\ y &= \frac {t \ln \left (t \right )+c_1 -t}{\operatorname {LambertW}\left (2 \left (t \ln \left (t \right )+c_1 -t \right ) {\mathrm e}^{-1}\right )} \\ \end{align*}

✓ Mathematica. Time used: 60.048 (sec). Leaf size: 76

ode=( 2*Log[t])-( Log[4*y[t]^2] )*D[y[t],t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {-2 t+2 t \log (t)+c_1}{2 W\left (-\frac {-2 t+2 t \log (t)+c_1}{e}\right )}\\ y(t)&\to \frac {-2 t+2 t \log (t)+c_1}{2 W\left (\frac {-2 t+2 t \log (t)+c_1}{e}\right )} \end{align*}

✓ Sympy. Time used: 0.409 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(2*log(t) - log(4*y(t)**2)*Derivative(y(t), t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {e^{W\left (C_{1} + \frac {2 t \log {\left (t \right )}}{e} - \frac {2 t}{e}\right ) + 1}}{2} \]