80.2.4 problem 5

Internal problem ID [21147]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 2. Theory of first order differential equations. Excercise 2.6 at page 37
Problem number : 5
Date solved : Thursday, October 02, 2025 at 07:10:16 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 20

ode:=diff(x(t),t)+sin(t)/(1+exp(t))*x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x = c_1 \,{\mathrm e}^{-\int \frac {\sin \left (t \right )}{1+{\mathrm e}^{t}}d t} \]

✓ Mathematica. Time used: 1.013 (sec). Leaf size: 74

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

\begin{align*} x(t)&\to c_1 \exp \left (\frac {1}{2} ((\cos (t)-i \sin (t)) \operatorname {Hypergeometric2F1}(-i,1,1-i,-\cosh (t)-\sinh (t))+(\cos (t)+i \sin (t)) \operatorname {Hypergeometric2F1}(i,1,1+i,-\cosh (t)-\sinh (t)))\right )\\ x(t)&\to 0 \end{align*}

✓ Sympy. Time used: 0.507 (sec). Leaf size: 15

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

\[ x{\left (t \right )} = C_{1} e^{\int \frac {\sin {\left (t \right )}}{e^{t} + 1}\, dt} \]