72.6.2 problem 2 (b)

Internal problem ID [19456]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Section 10 (Linear equations). Problems at page 82
Problem number : 2 (b)
Date solved : Thursday, October 02, 2025 at 04:28:31 PM
CAS classification : [_linear]

\begin{align*} y^{\prime }+y&=\frac {1}{{\mathrm e}^{2 x}+1} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 16

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

\[ y = \left (\frac {\arctan \left (\sinh \left (x \right )\right )}{2}+c_1 \right ) {\mathrm e}^{-x} \]

✓ Mathematica. Time used: 0.059 (sec). Leaf size: 18

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

\begin{align*} y(x)&\to e^{-x} \left (\arctan \left (e^x\right )+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.217 (sec). Leaf size: 27

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

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