50.3.3 problem 1(c)

Internal problem ID [7827]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.4 First Order Linear Equations. Page 15
Problem number : 1(c)
Date solved : Wednesday, March 05, 2025 at 05:07:21 AM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 14

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

\[ y = \left (\arctan \left ({\mathrm e}^{x}\right )+c_{1} \right ) {\mathrm e}^{-x} \]

✓ Mathematica. Time used: 0.067 (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]
 

\[ y(x)\to e^{-x} \left (\arctan \left (e^x\right )+c_1\right ) \]

✓ Sympy. Time used: 0.349 (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)
 

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