68.4.24 problem 24

Internal problem ID [17204]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number : 24
Date solved : Thursday, October 02, 2025 at 01:53:06 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.184 (sec). Leaf size: 22

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

\[ y = -\ln \left (2\right )-\ln \left (\frac {1}{\ln \left (t \right )^{2} c_1 -1}\right ) \]

✓ Mathematica. Time used: 0.403 (sec). Leaf size: 51

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

\begin{align*} y(t)&\to \log \left (\frac {1}{2} \left (-1+e^{2 c_1} \log ^2(t)\right )\right )\\ y(t)&\to -\log (2)-i \pi \\ y(t)&\to -\log (2)+i \pi \end{align*}

✓ Sympy. Time used: 0.180 (sec). Leaf size: 14

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

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