57.12.9 problem 7

Internal problem ID [14452]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.4.2 Variation of parameters. Exercises page 124
Problem number : 7
Date solved : Thursday, October 02, 2025 at 09:37:27 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 39

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

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

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 51

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

\begin{align*} x(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: 0.170 (sec). Leaf size: 36

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

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