68.18.22 problem 28

Internal problem ID [17866]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Chapter 4 review exercises, page 219
Problem number : 28
Date solved : Thursday, October 02, 2025 at 02:29:00 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-9 y&=\frac {1}{1+{\mathrm e}^{3 t}} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 67

ode:=diff(diff(y(t),t),t)-9*y(t) = 1/(1+exp(3*t)); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{3 t} c_2 +{\mathrm e}^{-3 t} c_1 +\frac {{\mathrm e}^{3 t} \left (-3 \ln \left ({\mathrm e}^{t}\right )+\ln \left ({\mathrm e}^{2 t}-{\mathrm e}^{t}+1\right )+\ln \left ({\mathrm e}^{t}+1\right )\right )}{18}-\frac {1}{18}+\frac {\left (\ln \left ({\mathrm e}^{-3 t}\right )-\ln \left (1+{\mathrm e}^{-3 t}\right )\right ) {\mathrm e}^{-3 t}}{18} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 63

ode=D[y[t],{t,2}]-9*y[t]==1/(1+Exp[3*t]); 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{9} e^{3 t} \text {arctanh}\left (2 e^{3 t}+1\right )-\frac {1}{18} e^{-3 t} \log \left (6 \left (e^{3 t}+1\right )\right )+c_1 e^{3 t}+c_2 e^{-3 t}-\frac {1}{18} \end{align*}

✓ Sympy. Time used: 0.208 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-9*y(t) + Derivative(y(t), (t, 2)) - 1/(exp(3*t) + 1),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} - \frac {\log {\left (e^{3 t} + 1 \right )}}{18}\right ) e^{- 3 t} + \left (C_{2} - \frac {t}{6} + \frac {\log {\left (e^{3 t} + 1 \right )}}{18}\right ) e^{3 t} - \frac {1}{18} \]