90.25.16 problem 16

Internal problem ID [25397]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 5. Second Order Linear Differential Equations. Exercises at page 379
Problem number : 16
Date solved : Friday, October 03, 2025 at 12:01:07 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 39

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

\[ y = {\mathrm e}^{t} c_2 +{\mathrm e}^{-t} c_1 +\frac {\left (-{\mathrm e}^{t}+{\mathrm e}^{-t}\right ) \ln \left ({\mathrm e}^{t}+1\right )}{2}+\frac {{\mathrm e}^{t} \ln \left ({\mathrm e}^{t}\right )}{2}-\frac {1}{2} \]

✓ Mathematica. Time used: 0.053 (sec). Leaf size: 51

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

\begin{align*} y(t)&\to -e^t \text {arctanh}\left (2 e^t+1\right )+\frac {1}{2} e^{-t} \log \left (e^t+1\right )+c_1 e^t+c_2 e^{-t}-\frac {1}{2} \end{align*}

✓ Sympy. Time used: 0.190 (sec). Leaf size: 36

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

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